cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A007318 Pascal's triangle read by rows: C(n,k) = binomial(n,k) = n!/(k!*(n-k)!), 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 10, 10, 5, 1, 1, 6, 15, 20, 15, 6, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1
Offset: 0

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Author

N. J. A. Sloane and Mira Bernstein, Apr 28 1994

Keywords

Comments

A. W. F. Edwards writes: "It [the triangle] was first written down long before 1654, the year in which Blaise Pascal wrote his Traité du triangle arithmétique, but it was this work that brought together all the different aspects of the numbers for the first time. In it Pascal developed the properties of the number as a piece of pure mathematics ... and then, in a series of appendices, showed how these properties were relevant to the study of the figurate numbers, to the theory of combinations, to the expansion of binomial expressions, and to the solution of an important problem in the theory of probability." (A. W. F. Edwards, Pascal's Arithmetical Triangle, Johns Hopkins University Press (2002), p. xiii)
Edwards reports that the naming of the triangle after Pascal was done first by Montmort in 1708 as the "Table de M. Pascal pour les combinaisons" and then by De Moivre in 1730 as the "Triangulum Arithmeticum PASCALANIUM". (Edwards, p. xiv)
In China, Yang Hui in 1261 listed the coefficients of (a+b)^n up to n=6, crediting the expansion to Chia Hsein's Shih-so suan-shu circa 1100. Another prominent early use was in Chu Shih-Chieh's Precious Mirror of the Four Elements in 1303. (Edwards, p. 51)
In Persia, Al-Karaji discovered the binomial triangle "some time soon after 1007", and Al-Samawal published it in the Al-bahir some time before 1180. (Edwards, p. 52)
In India, Halayuda's commentary (circa 900) on Pingala's treatise on syllabic combinations (circa 200 B.C.E.) contains a clear description of the additive computation of the triangle. (Amulya Kumar Bag, Binomial Theorem in Ancient India, p. 72)
Also in India, the multiplicative formula for C(n,k) was known to Mahavira in 850 and restated by Bhaskara in 1150. (Edwards, p. 27)
In Italy, Tartaglia published the triangle in his General trattato (1556), and Cardano published it in his Opus novum (1570). (Edwards, p. 39, 44) - Russ Cox, Mar 29 2022
Also sometimes called Omar Khayyam's triangle.
Also sometimes called Yang Hui's triangle.
C(n,k) = number of k-element subsets of an n-element set.
Row n gives coefficients in expansion of (1+x)^n.
Binomial(n+k-1,n-1) is the number of ways of placing k indistinguishable balls into n boxes (the "bars and stars" argument - see Feller).
Binomial(n-1,k-1) is the number of compositions (ordered partitions) of n with k summands.
Binomial(n+k-1,k-1) is the number of weak compositions (ordered weak partitions) of n into exactly k summands. - Juergen Will, Jan 23 2016
Binomial(n,k) is the number of lattice paths from (0,0) to (n,k) using steps (1,0) and (1,1). - Joerg Arndt, Jul 01 2011
If thought of as an infinite lower triangular matrix, inverse begins:
+1
-1 +1
+1 -2 +1
-1 +3 -3 +1
+1 -4 +6 -4 +1
All 2^n palindromic binomial coefficients starting after the A006516(n)-th entry are odd. - Lekraj Beedassy, May 20 2003
Binomial(n+k-1,n-1) is the number of standard tableaux of shape (n,1^k). - Emeric Deutsch, May 13 2004
Can be viewed as an array, read by antidiagonals, where the entries in the first row and column are all 1's and A(i,j) = A(i-1,j) + A(i,j-1) for all other entries. The determinant of each of its n X n subarrays starting at (0,0) is 1. - Gerald McGarvey, Aug 17 2004
Also the lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals j+1 (and all other entries are zero). - Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006
Binomial(n-3,k-1) counts the permutations in S_n which have zero occurrences of the pattern 231 and one occurrence of the pattern 132 and k descents. Binomial(n-3,k-1) also counts the permutations in S_n which have zero occurrences of the pattern 231 and one occurrence of the pattern 213 and k descents. - David Hoek (david.hok(AT)telia.com), Feb 28 2007
Inverse of A130595 (as an infinite lower triangular matrix). - Philippe Deléham, Aug 21 2007
Consider integer lists LL of lists L of the form LL = [m#L] = [m#[k#2]] (where '#' means 'times') like LL(m=3,k=3) = [[2,2,2],[2,2,2],[2,2,2]]. The number of the integer list partitions of LL(m,k) is equal to binomial(m+k,k) if multiple partitions like [[1,1],[2],[2]] and [[2],[2],[1,1]] and [[2],[1,1],[2]] are counted only once. For the example, we find 4*5*6/3! = 20 = binomial(6,3). - Thomas Wieder, Oct 03 2007
The infinitesimal generator for Pascal's triangle and its inverse is A132440. - Tom Copeland, Nov 15 2007
Row n>=2 gives the number of k-digit (k>0) base n numbers with strictly decreasing digits; e.g., row 10 for A009995. Similarly, row n-1>=2 gives the number of k-digit (k>1) base n numbers with strictly increasing digits; see A009993 and compare A118629. - Rick L. Shepherd, Nov 25 2007
From Lee Naish (lee(AT)cs.mu.oz.au), Mar 07 2008: (Start)
Binomial(n+k-1, k) is the number of ways a sequence of length k can be partitioned into n subsequences (see the Naish link).
Binomial(n+k-1, k) is also the number of n- (or fewer) digit numbers written in radix at least k whose digits sum to k. For example, in decimal, there are binomial(3+3-1,3)=10 3-digit numbers whose digits sum to 3 (see A052217) and also binomial(4+2-1,2)=10 4-digit numbers whose digits sum to 2 (see A052216). This relationship can be used to generate the numbers of sequences A052216 to A052224 (and further sequences using radix greater than 10). (End)
From Milan Janjic, May 07 2008: (Start)
Denote by sigma_k(x_1,x_2,...,x_n) the elementary symmetric polynomials. Then:
Binomial(2n+1,2k+1) = sigma_{n-k}(x_1,x_2,...,x_n), where x_i = tan^2(i*Pi/(2n+1)), (i=1,2,...,n).
Binomial(2n,2k+1) = 2n*sigma_{n-1-k}(x_1,x_2,...,x_{n-1}), where x_i = tan^2(i*Pi/(2n)), (i=1,2,...,n-1).
Binomial(2n,2k) = sigma_{n-k}(x_1,x_2,...,x_n), where x_i = tan^2((2i-1)Pi/(4n)), (i=1,2,...,n).
Binomial(2n+1,2k) = (2n+1)sigma_{n-k}(x_1,x_2,...,x_n), where x_i = tan^2((2i-1)Pi/(4n+2)), (i=1,2,...,n). (End)
Given matrices R and S with R(n,k) = binomial(n,k)*r(n-k) and S(n,k) = binomial(n,k)*s(n-k), then R*S = T where T(n,k) = binomial(n,k)*[r(.)+s(.)]^(n-k), umbrally. And, the e.g.f.s for the row polynomials of R, S and T are, respectively, exp(x*t)*exp[r(.)*x], exp(x*t)*exp[s(.)*x] and exp(x*t)*exp[r(.)*x]*exp[s(.)*x] = exp{[t+r(.)+s(.)]*x}. The row polynomials are essentially Appell polynomials. See A132382 for an example. - Tom Copeland, Aug 21 2008
As the rectangle R(m,n) = binomial(m+n-2,m-1), the weight array W (defined generally at A144112) of R is essentially R itself, in the sense that if row 1 and column 1 of W=A144225 are deleted, the remaining array is R. - Clark Kimberling, Sep 15 2008
If A007318 = M as an infinite lower triangular matrix, M^n gives A130595, A023531, A007318, A038207, A027465, A038231, A038243, A038255, A027466, A038279, A038291, A038303, A038315, A038327, A133371, A147716, A027467 for n=-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 respectively. - Philippe Deléham, Nov 11 2008
The coefficients of the polynomials with e.g.f. exp(x*t)*(cosh(t)+sinh(t)). - Peter Luschny, Jul 09 2009
The triangle or chess sums, see A180662 for their definitions, link Pascal's triangle with twenty different sequences, see the crossrefs. All sums come in pairs due to the symmetrical nature of this triangle. The knight sums Kn14 - Kn110 have been added. It is remarkable that all knight sums are related to the Fibonacci numbers, i.e., A000045, but none of the others. - Johannes W. Meijer, Sep 22 2010
Binomial(n,k) is also the number of ways to distribute n+1 balls into k+1 urns so that each urn gets at least one ball. See example in the example section below. - Dennis P. Walsh, Jan 29 2011
Binomial(n,k) is the number of increasing functions from {1,...,k} to {1,...,n} since there are binomial(n,k) ways to choose the k distinct, ordered elements of the range from the codomain {1,...,n}. See example in the example section below. - Dennis P. Walsh, Apr 07 2011
Central binomial coefficients: T(2*n,n) = A000984(n), T(n, floor(n/2)) = A001405(n). - Reinhard Zumkeller, Nov 09 2011
Binomial(n,k) is the number of subsets of {1,...,n+1} with k+1 as median element. To see this, note that Sum_{j=0..min(k,n-k)}binomial(k,j)*binomial(n-k,j) = binomial(n,k). See example in Example section below. - Dennis P. Walsh, Dec 15 2011
This is the coordinator triangle for the lattice Z^n, see Conway-Sloane, 1997. - N. J. A. Sloane, Jan 17 2012
One of three infinite families of integral factorial ratio sequences of height 1 (see Bober, Theorem 1.2). The other two are A046521 and A068555. For real r >= 0, C_r(n,k) := floor(r*n)!/(floor(r*k)!*floor(r*(n-k))!) is integral. See A211226 for the case r = 1/2. - Peter Bala, Apr 10 2012
Define a finite triangle T(m,k) with n rows such that T(m,0) = 1 is the left column, T(m,m) = binomial(n-1,m) is the right column, and the other entries are T(m,k) = T(m-1,k-1) + T(m-1,k) as in Pascal's triangle. The sum of all entries in T (there are A000217(n) elements) is 3^(n-1). - J. M. Bergot, Oct 01 2012
The lower triangular Pascal matrix serves as a representation of the operator exp(RLR) in a basis composed of a sequence of polynomials p_n(x) characterized by ladder operators defined by R p_n(x) = p_(n+1)(x) and L p_n(x) = n p_(n-1)(x). See A132440, A218272, A218234, A097805, and A038207. The transposed and padded Pascal matrices can be associated to the special linear group SL2. - Tom Copeland, Oct 25 2012
See A193242. - Alexander R. Povolotsky, Feb 05 2013
A permutation p_1...p_n of the set {1,...,n} has a descent at position i if p_i > p_(i+1). Let S(n) denote the subset of permutations p_1...p_n of {1,...,n} such that p_(i+1) - p_i <= 1 for i = 1,...,n-1. Then binomial(n,k) gives the number of permutations in S(n+1) with k descents. Alternatively, binomial(n,k) gives the number of permutations in S(n+1) with k+1 increasing runs. - Peter Bala, Mar 24 2013
Sum_{n=>0} binomial(n,k)/n! = e/k!, where e = exp(1), while allowing n < k where binomial(n,k) = 0. Also Sum_{n>=0} binomial(n+k-1,k)/n! = e * A000262(k)/k!, and for k>=1 equals e * A067764(k)/A067653(k). - Richard R. Forberg, Jan 01 2014
The square n X n submatrix (first n rows and n columns) of the Pascal matrix P(x) defined in the formulas below when multiplying on the left the Vandermonde matrix V(x_1,...,x_n) (with ones in the first row) translates the matrix to V(x_1+x,...,x_n+x) while leaving the determinant invariant. - Tom Copeland, May 19 2014
For k>=2, n>=k, k/((k/(k-1) - Sum_{n=k..m} 1/binomial(n,k))) = m!/((m-k+1)!*(k-2)!). Note: k/(k-1) is the infinite sum. See A000217, A000292, A000332 for examples. - Richard R. Forberg, Aug 12 2014
Let G_(2n) be the subgroup of the symmetric group S_(2n) defined by G_(2n) = {p in S_(2n) | p(i) = i (mod n) for i = 1,2,...,2n}. G_(2n) has order 2^n. Binomial(n,k) gives the number of permutations in G_(2n) having n + k cycles. Cf. A130534 and A246117. - Peter Bala, Aug 15 2014
C(n,k) = the number of Dyck paths of semilength n+1, with k+1 "u"'s in odd numbered positions and k+1 returns to the x axis. Example: {U = u in odd position and = return to x axis} binomial(3,0)=1 (Uudududd); binomial(3,1)=3 [(Uududd_Ud_), (Ud_Uududd_), (Uudd_Uudd_)]; binomial(3,2)=3 [(Ud_Ud_Uudd_), (Uudd_Ud_Ud_), (Ud_Uudd_Ud_)]; binomial(3,3)=1 (Ud_Ud_Ud_Ud_). - Roger Ford, Nov 05 2014
From Daniel Forgues, Mar 12 2015: (Start)
The binomial coefficients binomial(n,k) give the number of individuals of the k-th generation after n population doublings. For each doubling of population, each individual's clone has its generation index incremented by 1, and thus goes to the next row. Just tally up each row from 0 to 2^n - 1 to get the binomial coefficients.
0 1 3 7 15
0: O | . | . . | . . . . | . . . . . . . . |
1: | O | O . | O . . . | O . . . . . . . |
2: | | O | O O . | O O . O . . . |
3: | | | O | O O O . |
4: | | | | O |
This is a fractal process: to get the pattern from 0 to 2^n - 1, append a shifted down (by one row) copy of the pattern from 0 to 2^(n-1) - 1 to the right of the pattern from 0 to 2^(n-1) - 1. (Inspired by the "binomial heap" data structure.)
Sequence of generation indices: 1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n) (see A000120):
{0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, ...}
Binary expansion of 0 to 15:
0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1111
(End)
A258993(n,k) = T(n+k,n-k), n > 0. - Reinhard Zumkeller, Jun 22 2015
T(n,k) is the number of set partitions w of [n+1] that avoid 1/2/3 with rb(w)=k. The same holds for ls(w)=k, where avoidance is in the sense of Klazar and ls,rb defined by Wachs and White.
Satisfies Benford's law [Diaconis, 1977] - N. J. A. Sloane, Feb 09 2017
Let {A(n)} be a set with exactly n identical elements, with {A(0)} being the empty set E. Let {A(n,k)} be the k-th iteration of {A(n)}, with {A(n,0)} = {A(n)}. {A(n,1)} = The set of all the subsets of A{(n)}, including {A(n)} and E. {A(n,k)} = The set of all subsets of {A(n,k-1)}, including all of the elements of {A(n,k-1)}. Let A(n,k) be the number of elements in {A(n,k)}. Then A(n,k) = C(n+k,k), with each successive iteration replicating the members of the k-th diagonal of Pascal's Triangle. See examples. - Gregory L. Simay, Aug 06 2018
Binomial(n-1,k) is also the number of permutations avoiding both 213 and 312 with k ascents. - Lara Pudwell, Dec 19 2018
Binomial(n-1,k) is also the number of permutations avoiding both 132 and 213 with k ascents. - Lara Pudwell, Dec 19 2018
Binomial(n,k) is the dimension of the k-th exterior power of a vector space of dimension n. - Stefano Spezia, Dec 22 2018
C(n,k-1) is the number of unoriented colorings of the facets (or vertices) of an n-dimensional simplex using exactly k colors. Each chiral pair is counted as one when enumerating unoriented arrangements. - Robert A. Russell, Oct 20 2020
From Dilcher and Stolarsky: "Two of the most ubiquitous objects in mathematics are the sequence of prime numbers and the binomial coefficients (and thus Pascal's triangle). A connection between the two is given by a well-known characterization of the prime numbers: Consider the entries in the k-th row of Pascal's triangle, without the initial and final entries. They are all divisible by k if and only if k is a prime." - Tom Copeland, May 17 2021
Named "Table de M. Pascal pour les combinaisons" by Pierre Remond de Montmort (1708) after the French mathematician, physicist and philosopher Blaise Pascal (1623-1662). - Amiram Eldar, Jun 11 2021
Consider the n-th diagonal of the triangle as a sequence b(n) with n starting at 0. From it form a new sequence by leaving the 0th term as is, and thereafter considering all compositions of n, taking the product of b(i) over the respective numbers i in each composition, adding terms corresponding to compositions with an even number of parts subtracting terms corresponding to compositions with an odd number of parts. Then the n-th row of the triangle is obtained, with every second term multiplied by -1, followed by infinitely many zeros. For sequences starting with 1, this operation is a special case of a self-inverse operation, and therefore the converse is true. - Thomas Anton, Jul 05 2021
C(n,k) is the number of vertices in an n-dimensional unit hypercube, at an L1 distance of k (or: with a shortest path of k 1d-edges) from a given vertex. - Eitan Y. Levine, May 01 2023
C(n+k-1,k-1) is the number of vertices at an L1 distance from a given vertex in an infinite-dimensional box, which has k sides of length 2^m for each m >= 0. Equivalently, given a set of tokens containing k distinguishable tokens with value 2^m for each m >= 0, C(n+k-1,k-1) is the number of subsets of tokens with a total value of n. - Eitan Y. Levine, Jun 11 2023
Numbers in the k-th column, i.e., numbers of the form C(n,k) for n >= k, are known as k-simplex numbers. - Pontus von Brömssen, Jun 26 2023
Let r(k) be the k-th row and c(k) the k-th column. Denote convolution by * and repeated convolution by ^. Then r(k)*r(m)=r(k+m) and c(k)*c(m)=c(k+m+1). This is because r(k) = r(1) ^ k and c(k) = c(0) ^ k+1. - Eitan Y. Levine, Jul 23 2023
Number of permutations of length n avoiding simultaneously the patterns 231 and 312(resp., 213 and 231; 213 and 312) with k descents (equivalently, with k ascents). An ascent (resp., descent) in a permutation a(1)a(2)...a(n) is position i such that a(i)a(i+1)). - Tian Han, Nov 25 2023
C(n,k) are generalized binomial coefficients of order m=0. Calculated by the formula C(n,k) = Sum_{i=0..n-k} binomial(n+1, n-k-i)*Stirling2(i+ m+ 1, i+1) *(-1)^i, where m = 0 for n>= 0, 0 <= k <= n. - Igor Victorovich Statsenko, Feb 26 2023
The Akiyama-Tanigawa algorithm applied to the diagonals, binomial(n+k,k), yields the powers of n. - Shel Kaphan, May 03 2024

Examples

			Triangle T(n,k) begins:
   n\k 0   1   2   3   4   5   6   7   8   9  10  11 ...
   0   1
   1   1   1
   2   1   2   1
   3   1   3   3   1
   4   1   4   6   4   1
   5   1   5  10  10   5   1
   6   1   6  15  20  15   6   1
   7   1   7  21  35  35  21   7   1
   8   1   8  28  56  70  56  28   8   1
   9   1   9  36  84 126 126  84  36   9   1
  10   1  10  45 120 210 252 210 120  45  10   1
  11   1  11  55 165 330 462 462 330 165  55  11   1
  ...
There are C(4,2)=6 ways to distribute 5 balls BBBBB, among 3 different urns, < > ( ) [ ], so that each urn gets at least one ball, namely, <BBB>(B)[B], <B>(BBB)[B], <B>(B)[BBB], <BB>(BB)[B], <BB>(B)[BB], and <B>(BB)[BB].
There are C(4,2)=6 increasing functions from {1,2} to {1,2,3,4}, namely, {(1,1),(2,2)},{(1,1),(2,3)}, {(1,1),(2,4)}, {(1,2),(2,3)}, {(1,2),(2,4)}, and {(1,3),(2,4)}. - _Dennis P. Walsh_, Apr 07 2011
There are C(4,2)=6 subsets of {1,2,3,4,5} with median element 3, namely, {3}, {1,3,4}, {1,3,5}, {2,3,4}, {2,3,5}, and {1,2,3,4,5}. - _Dennis P. Walsh_, Dec 15 2011
The successive k-iterations of {A(0)} = E are E;E;E;...; the corresponding number of elements are 1,1,1,... The successive k-iterations of {A(1)} = {a} are (omitting brackets) a;a,E; a,E,E;...; the corresponding number of elements are 1,2,3,... The successive k-iterations of {A(2)} = {a,a} are aa; aa,a,E; aa, a, E and a,E and E;...; the corresponding number of elements are 1,3,6,... - _Gregory L. Simay_, Aug 06 2018
Boas-Buck type recurrence for column k = 4: T(8, 4) = (5/4)*(1 + 5 + 15 + 35) = 70. See the Boas-Buck comment above. - _Wolfdieter Lang_, Nov 12 2018

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 828.
  • Amulya Kumar Bag, Binomial theorem in ancient India, Indian Journal of History of Science, vol. 1 (1966), pp. 68-74.
  • Arthur T. Benjamin and Jennifer Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 63ff.
  • Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
  • Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 306.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 68-74.
  • Paul Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
  • A. W. F. Edwards, Pascal's Arithmetical Triangle, 2002.
  • William Feller, An Introduction to Probability Theory and Its Application, Vol. 1, 2nd ed. New York: Wiley, p. 36, 1968.
  • Ronald L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd. ed., 1994, p. 155.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, pp. 140-141.
  • David Hök, Parvisa mönster i permutationer [Swedish], 2007.
  • Donald E. Knuth, The Art of Computer Programming, Vol. 1, 2nd ed., p. 52.
  • Sergei K. Lando, Lecture on Generating Functions, Amer. Math. Soc., Providence, R.I., 2003, pp. 60-61.
  • Blaise Pascal, Traité du triangle arithmétique, avec quelques autres petits traitez sur la mesme matière, Desprez, Paris, 1665.
  • Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 71.
  • Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 271-275.
  • A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
  • John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 6.
  • John Riordan, Combinatorial Identities, Wiley, 1968, p. 2.
  • Robert Sedgewick and Philippe Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996, p. 143.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 6, pages 43-52.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 13, 30-33.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 115-118.
  • Douglas B. West, Combinatorial Mathematics, Cambridge, 2021, p. 25.

Links

Crossrefs

Equals differences between consecutive terms of A102363. - David G. Williams (davidwilliams(AT)Paxway.com), Jan 23 2006
Row sums give A000079 (powers of 2).
Cf. A083093 (triangle read mod 3), A214292 (first differences of rows).
Partial sums of rows give triangle A008949.
The triangle of the antidiagonals is A011973.
Infinite matrix squared: A038207, cubed: A027465.
Cf. A101164. If rows are sorted we get A061554 or A107430.
Another version: A108044.
Cf. A008277, A132311, A132312, A052216, A052217, A052218, A052219, A052220, A052221, A052222, A052223, A144225, A202750, A211226, A047999, A026729, A052553, A051920, A193242.
Triangle sums (see the comments): A000079 (Row1); A000007 (Row2); A000045 (Kn11 & Kn21); A000071 (Kn12 & Kn22); A001924 (Kn13 & Kn23); A014162 (Kn14 & Kn24); A014166 (Kn15 & Kn25); A053739 (Kn16 & Kn26); A053295 (Kn17 & Kn27); A053296 (Kn18 & Kn28); A053308 (Kn19 & Kn29); A053309 (Kn110 & Kn210); A001519 (Kn3 & Kn4); A011782 (Fi1 & Fi2); A000930 (Ca1 & Ca2); A052544 (Ca3 & Ca4); A003269 (Gi1 & Gi2); A055988 (Gi3 & Gi4); A034943 (Ze1 & Ze2); A005251 (Ze3 & Ze4). - Johannes W. Meijer, Sep 22 2010
Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A105809, A109906, A111006, A114197, A162741, A228074, A228196, A228576.
Cf. A137948, A245334.
Cf. A085478, A258993.
Cf. A115940 (pandigital binomial coefficients C(m,k) with k>1).
Cf. (simplex colorings) A325002 (oriented), [k==n+1] (chiral), A325003 (achiral), A325000 (k or fewer colors), A325009 (orthotope facets, orthoplex vertices), A325017 (orthoplex facets, orthotope vertices).
Triangles of generalized binomial coefficients (n,k)_m (or generalized Pascal triangles) for m = 2..12: A001263, A056939, A056940, A056941, A142465, A142467, A142468, A174109, A342889, A342890, A342891.

Programs

  • Axiom
    -- (start)
)set expose add constructor OutputForm
pascal(0,n) == 1
pascal(n,n) == 1
pascal(i,j | 0 < i and i < j) == pascal(i-1,j-1) + pascal(i,j-1)
pascalRow(n) == [pascal(i,n) for i in 0..n]
displayRow(n) == output center blankSeparate pascalRow(n)
for i in 0..20 repeat displayRow i -- (end)
  • GAP
    Flat(List([0..12],n->List([0..n],k->Binomial(n,k)))); # Stefano Spezia, Dec 22 2018
  • Haskell
    a007318 n k = a007318_tabl !! n !! k
a007318_row n = a007318_tabl !! n
a007318_list = concat a007318_tabl
a007318_tabl = iterate (\row -> zipWith (+) ([0] ++ row) (row ++ [0])) [1]
-- Cf. http://www.haskell.org/haskellwiki/Blow_your_mind#Mathematical_sequences
-- Reinhard Zumkeller, Nov 09 2011, Oct 22 2010
  • Magma
    /* As triangle: */ [[Binomial(n, k): k in [0..n]]: n in [0.. 10]]; // Vincenzo Librandi, Jul 29 2015
  • Maple
    A007318 := (n,k)->binomial(n,k);
  • Mathematica
    Flatten[Table[Binomial[n, k], {n, 0, 11}, {k, 0, n}]] (* Robert G. Wilson v, Jan 19 2004 *)
Flatten[CoefficientList[CoefficientList[Series[1/(1 - x - x*y), {x, 0, 12}], x], y]] (* Mats Granvik, Jul 08 2014 *)
  • Maxima
    create_list(binomial(n,k),n,0,12,k,0,n); /* Emanuele Munarini, Mar 11 2011 */
  • PARI
    C(n,k)=binomial(n,k) \\ Charles R Greathouse IV, Jun 08 2011
  • Python
    # See Hobson link. Further programs:
from math import prod,factorial
def C(n,k): return prod(range(n,n-k,-1))//factorial(k) # M. F. Hasler, Dec 13 2019, updated Apr 29 2022, Feb 17 2023
  • Python
    from math import comb, isqrt
def A007318(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),n-comb(r+1,2)) # Chai Wah Wu, Nov 11 2024
  • Sage
    def C(n,k): return Subsets(range(n), k).cardinality() # Ralf Stephan, Jan 21 2014

Formula

a(n, k) = C(n,k) = binomial(n, k).
C(n, k) = C(n-1, k) + C(n-1, k-1).
The triangle is symmetric: C(n,k) = C(n,n-k).
a(n+1, m) = a(n, m) + a(n, m-1), a(n, -1) := 0, a(n, m) := 0, n
C(n, k) = n!/(k!(n-k)!) if 0<=k<=n, otherwise 0.
C(n, k) = ((n-k+1)/k) * C(n, k-1) with C(n, 0) = 1. - Michael B. Porter, Mar 23 2025
G.f.: 1/(1-y-x*y) = Sum_(C(n, k)*x^k*y^n, n, k>=0)
G.f.: 1/(1-x-y) = Sum_(C(n+k, k)*x^k*y^n, n, k>=0).
G.f. for row n: (1+x)^n = Sum_{k=0..n} C(n, k)*x^k.
G.f. for column k: x^k/(1-x)^(k+1); [corrected by Werner Schulte, Jun 15 2022].
E.g.f.: A(x, y) = exp(x+x*y).
E.g.f. for column n: x^n*exp(x)/n!.
In general the m-th power of A007318 is given by: T(0, 0) = 1, T(n, k) = T(n-1, k-1) + m*T(n-1, k), where n is the row-index and k is the column; also T(n, k) = m^(n-k)*C(n, k).
Triangle T(n, k) read by rows; given by A000007 DELTA A000007, where DELTA is Deléham's operator defined in A084938.
Let P(n+1) = the number of integer partitions of (n+1); let p(i) = the number of parts of the i-th partition of (n+1); let d(i) = the number of different parts of the i-th partition of (n+1); let m(i, j) = multiplicity of the j-th part of the i-th partition of (n+1). Define the operator Sum_{i=1..P(n+1), p(i)=k+1} as the sum running from i=1 to i=P(n+1) but taking only partitions with p(i)=(k+1) parts into account. Define the operator Product_{j=1..d(i)} = product running from j=1 to j=d(i). Then C(n, k) = Sum_{p(i)=(k+1), i=1..P(n+1)} p(i)! / [Product_{j=1..d(i)} m(i, j)!]. E.g., C(5, 3) = 10 because n=6 has the following partitions with m=3 parts: (114), (123), (222). For their multiplicities one has: (114): 3!/(2!*1!) = 3; (123): 3!/(1!*1!*1!) = 6; (222): 3!/3! = 1. The sum is 3 + 6 + 1 = 10 = C(5, 3). - Thomas Wieder, Jun 03 2005
C(n, k) = Sum_{j=0..k} (-1)^j*C(n+1+j, k-j)*A000108(j). - Philippe Deléham, Oct 10 2005
G.f.: 1 + x*(1 + x) + x^3*(1 + x)^2 + x^6*(1 + x)^3 + ... . - Michael Somos, Sep 16 2006
Sum_{k=0..floor(n/2)} x^(n-k)*T(n-k,k) = A000007(n), A000045(n+1), A002605(n), A030195(n+1), A057087(n), A057088(n), A057089(n), A057090(n), A057091(n), A057092(n), A057093(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, respectively. Sum_{k=0..floor(n/2)} (-1)^k*x^(n-k)*T(n-k,k) = A000007(n), A010892(n), A009545(n+1), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n+1), A057086(n), A084329(n+1) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, respectively. - Philippe Deléham, Sep 16 2006
C(n,k) <= A062758(n) for n > 1. - Reinhard Zumkeller, Mar 04 2008
C(t+p-1, t) = Sum_{i=0..t} C(i+p-2, i) = Sum_{i=1..p} C(i+t-2, t-1). A binomial number is the sum of its left parent and all its right ancestors, which equals the sum of its right parent and all its left ancestors. - Lee Naish (lee(AT)cs.mu.oz.au), Mar 07 2008
From Paul D. Hanna, Mar 24 2011: (Start)
Let A(x) = Sum_{n>=0} x^(n*(n+1)/2)*(1+x)^n be the g.f. of the flattened triangle:
A(x) = 1 + (x + x^2) + (x^3 + 2*x^4 + x^5) + (x^6 + 3*x^7 + 3*x^8 + x^9) + ...
then A(x) equals the series Sum_{n>=0} (1+x)^n*x^n*Product_{k=1..n} (1-(1+x)*x^(2*k-1))/(1-(1+x)*x^(2*k));
also, A(x) equals the continued fraction 1/(1- x*(1+x)/(1+ x*(1-x)*(1+x)/(1- x^3*(1+x)/(1+ x^2*(1-x^2)*(1+x)/(1- x^5*(1+x)/(1+ x^3*(1-x^3)*(1+x)/(1- x^7*(1+x)/(1+ x^4*(1-x^4)*(1+x)/(1- ...))))))))).
These formulas are due to (1) a q-series identity and (2) a partial elliptic theta function expression. (End)
For n > 0: T(n,k) = A029600(n,k) - A029635(n,k), 0 <= k <= n. - Reinhard Zumkeller, Apr 16 2012
Row n of the triangle is the result of applying the ConvOffs transform to the first n terms of the natural numbers (1, 2, 3, ..., n). See A001263 or A214281 for a definition of this transformation. - Gary W. Adamson, Jul 12 2012
From L. Edson Jeffery, Aug 02 2012: (Start)
Row n (n >= 0) of the triangle is given by the n-th antidiagonal of the infinite matrix P^n, where P = (p_{i,j}), i,j >= 0, is the production matrix
0, 1,
1, 0, 1,
0, 1, 0, 1,
0, 0, 1, 0, 1,
0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 1, 0, 1,
0, 0, 0, 0, 0, 0, 1, 0, 1,
... (End)
Row n of the triangle is also given by the n+1 coefficients of the polynomial P_n(x) defined by the recurrence P_0(x) = 1, P_1(x) = x + 1, P_n(x) = x*P_{n-1}(x) + P_{n-2}(x), n > 1. - L. Edson Jeffery, Aug 12 2013
For a closed-form formula for arbitrary left and right borders of Pascal-like triangles see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 04 2013
(1+x)^n = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*Sum_{i=0..k} k^(n-i)*binomial(k,i)*x^(n-i)/(n-i)!. - Vladimir Kruchinin, Oct 21 2013
E.g.f.: A(x,y) = exp(x+x*y) = 1 + (x+y*x)/( E(0)-(x+y*x)), where E(k) = 1 + (x+y*x)/(1 + (k+1)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 08 2013
E.g.f.: E(0) -1, where E(k) = 2 + x*(1+y)/(2*k+1 - x*(1+y)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013
G.f.: 1 + x*(1+x)*(1+x^2*(1+x)/(W(0)-x^2-x^3)), where W(k) = 1 + (1+x)*x^(k+2) - (1+x)*x^(k+3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Dec 24 2013
Sum_{n>=0} C(n,k)/n! = e/k!, where e = exp(1), while allowing n < k where C(n,k) = 0. Also Sum_{n>=0} C(n+k-1,k)/n! = e * A000262(k)/k!, and for k>=1 equals e * A067764(k)/A067653(k). - Richard R. Forberg, Jan 01 2014
Sum_{n>=k} 1/C(n,k) = k/(k-1) for k>=1. - Richard R. Forberg, Feb 10 2014
From Tom Copeland, Apr 26 2014: (Start)
Multiply each n-th diagonal of the Pascal lower triangular matrix by x^n and designate the result by A007318(x) = P(x). Then with :xD:^n = x^n*(d/dx)^n and B(n,x), the Bell polynomials (A008277),
A) P(x)= exp(x*dP) = exp[x*(e^M-I)] = exp[M*B(.,x)] = (I+dP)^B(.,x)
with dP = A132440, M = A238385-I, and I = identity matrix, and
B) P(:xD:) = exp(dP:xD:) = exp[(e^M-I):xD:] = exp[M*B(.,:xD:)] = exp[M*xD] = (I+dP)^(xD) with action P(:xD:)g(x) = exp(dP:xD:)g(x) = g[(I+dP)*x] (cf. also A238363).
C) P(x)^y = P(y*x). P(2x) = A038207(x) = exp[M*B(.,2x)], the face vectors of the n-dim hypercubes.
D) P(x) = [St2]*exp(x*M)*[St1] = [St2]*(I+dP)^x*[St1]
E) = [St1]^(-1)*(I+dP)^x*[St1] = [St2]*(I+dP)^x*[St2]^(-1)
where [St1]=padded A008275 just as [St2]=A048993=padded A008277 and exp(x*M) = (I+dP)^x = Sum_{k>=0} C(x,k) dP^k. (End)
T(n,k) = A245334(n,k) / A137948(n,k), 0 <= k <= n. - Reinhard Zumkeller, Aug 31 2014
From Peter Bala, Dec 21 2014: (Start)
Recurrence equation: T(n,k) = T(n-1,k)*(n + k)/(n - k) - T(n-1,k-1) for n >= 2 and 1 <= k < n, with boundary conditions T(n,0) = T(n,n) = 1. Note, changing the minus sign in the recurrence to a plus sign gives a recurrence for the square of the binomial coefficients - see A008459.
There is a relation between the e.g.f.'s of the rows and the diagonals of the triangle, namely, exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(1 + 3*x + 3*x^2/2! + x^3/3!) = 1 + 4*x + 10*x^2/2! + 20*x^3/3! + 35*x^4/4! + .... This property holds more generally for the Riordan arrays of the form ( f(x), x/(1 - x) ), where f(x) is an o.g.f. of the form 1 + f_1*x + f_2*x^2 + .... See, for example, A055248 and A106516.
Let P denote the present triangle. For k = 0,1,2,... define P(k) to be the lower unit triangular block array
/I_k 0\
\ 0 P/ having the k X k identity matrix I_k as the upper left block; in particular, P(0) = P. The infinite product P(0)*P(1)*P(2)*..., which is clearly well-defined, is equal to the triangle of Stirling numbers of the second kind A008277. The infinite product in the reverse order, that is, ...*P(2)*P(1)*P(0), is equal to the triangle of Stirling cycle numbers A130534. (End)
C(a+b,c) = Sum_{k=0..a} C(a,k)*C(b,b-c+k). This is a generalization of equation 1 from section 4.2.5 of the Prudnikov et al. reference, for a=b=c=n: C(2*n,n) = Sum_{k=0..n} C(n,k)^2. See Links section for animation of new formula. - Hermann Stamm-Wilbrandt, Aug 26 2015
The row polynomials of the Pascal matrix P(n,x) = (1+x)^n are related to the Bernoulli polynomials Br(n,x) and their umbral compositional inverses Bv(n,x) by the umbral relation P(n,x) = (-Br(.,-Bv(.,x)))^n = (-1)^n Br(n,-Bv(.,x)), which translates into the matrix relation P = M * Br * M * Bv, where P is the Pascal matrix, M is the diagonal matrix diag(1,-1,1,-1,...), Br is the matrix for the coefficients of the Bernoulli polynomials, and Bv that for the umbral inverse polynomials defined umbrally by Br(n,Bv(.,x)) = x^n = Bv(n,Br(.,x)). Note M = M^(-1). - Tom Copeland, Sep 05 2015
1/(1-x)^k = (r(x) * r(x^2) * r(x^4) * ...) where r(x) = (1+x)^k. - Gary W. Adamson, Oct 17 2016
Boas-Buck type recurrence for column k for Riordan arrays (see the Aug 10 2017 remark in A046521, also for the reference) with the Boas-Buck sequence b(n) = {repeat(1)}. T(n, k) = ((k+1)/(n-k))*Sum_{j=k..n-1} T(j, k), for n >= 1, with T(n, n) = 1. This reduces, with T(n, k) = binomial(n, k), to a known binomial identity (e.g, Graham et al. p. 161). - Wolfdieter Lang, Nov 12 2018
C((p-1)/a, b) == (-1)^b * fact_a(a*b-a+1)/fact_a(a*b) (mod p), where fact_n denotes the n-th multifactorial, a divides p-1, and the denominator of the fraction on the right side of the equation represents the modular inverse. - Isaac Saffold, Jan 07 2019
C(n,k-1) = A325002(n,k) - [k==n+1] = (A325002(n,k) + A325003(n,k)) / 2 = [k==n+1] + A325003(n,k). - Robert A. Russell, Oct 20 2020
From Hermann Stamm-Wilbrandt, May 13 2021: (Start)
Binomial sums are Fibonacci numbers A000045:
Sum_{k=0..n} C(n + k, 2*k + 1) = F(2*n).
Sum_{k=0..n} C(n + k, 2*k) = F(2*n + 1). (End)
C(n,k) = Sum_{i=0..k} A000108(i) * C(n-2i-1, k-i), for 0 <= k <= floor(n/2)-1. - Tushar Bansal, May 17 2025

Extensions

Checked all links, deleted 8 that seemed lost forever and were probably not of great importance. - N. J. A. Sloane, May 08 2018

A000032 Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.

Original entry on oeis.org

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803
Offset: 0

Views

Author

N. J. A. Sloane, May 24 1994

Keywords

Comments

Cf. A000204 for Lucas numbers beginning with 1.
Also the number of independent vertex sets and vertex covers for the cycle graph C_n for n >= 2. - Eric W. Weisstein, Jan 04 2014
Also the number of matchings in the n-cycle graph C_n for n >= 3. - Eric W. Weisstein, Oct 01 2017
Also the number of maximal independent vertex sets (and maximal vertex covers) for the n-helm graph for n >= 3. - Eric W. Weisstein, May 27 2017
Also the number of maximal independent vertex sets (and maximal vertex covers) for the n-sunlet graph for n >= 3. - Eric W. Weisstein, Aug 07 2017
This is also the Horadam sequence (2, 1, 1, 1). - Ross La Haye, Aug 18 2003
For distinct primes p, q, L(p) is congruent to 1 mod p, L(2p) is congruent to 3 mod p and L(pq) is congruent 1 + q(L(q) - 1) mod p. Also, L(m) divides F(2km) and L((2k + 1)m), k, m >= 0.
a(n) = Sum_{k=0..ceiling((n - 1)/2)} P(3; n - 1 - k, k), n >= 1, with a(0) = 2. These are the sums over the SW-NE diagonals in P(3; n, k), the (3, 1) Pascal triangle A093560. Observation by Paul Barry, Apr 29 2004. Proof via recursion relations and comparison of inputs. Also SW-NE diagonal sums of the (1, 2) Pascal triangle A029635 (with T(0, 0) replaced by 2).
Suppose psi = log(phi) = A002390. We get the representation L(n) = 2*cosh(n*psi) if n is even; L(n) = 2*sinh(n*psi) if n is odd. There is a similar representation for Fibonacci numbers (A000045). Many Lucas formulas now easily follow from appropriate sinh- and cosh-formulas. For example: the identity cosh^2(x) - sinh^2(x) = 1 implies L(n)^2 - 5*F(n)^2 = 4*(-1)^n (setting x = n*psi). - Hieronymus Fischer, Apr 18 2007
From John Blythe Dobson, Oct 02 2007, Oct 11 2007: (Start)
The parity of L(n) follows easily from its definition, which shows that L(n) is even when n is a multiple of 3 and odd otherwise.
The first six multiplication formulas are:
L(2n) = L(n)^2 - 2*(-1)^n;
L(3n) = L(n)^3 - 3*(-1)^n*L(n);
L(4n) = L(n)^4 - 4*(-1)^n*L(n)^2 + 2;
L(5n) = L(n)^5 - 5*(-1)^n*L(n)^3 + 5*L(n);
L(6n) = L(n)^6 - 6*(-1)^n*L(n)^4 + 9*L(n)^2 - 2*(-1)^n.
Generally, L(n) | L(mn) if and only if m is odd.
In the expansion of L(mn), where m represents the multiplier and n the index of a known value of L(n), the absolute values of the coefficients are the terms in the m-th row of the triangle A034807. When m = 1 and n = 1, L(n) = 1 and all the terms are positive and so the row sums of A034807 are simply the Lucas numbers. (End)
From John Blythe Dobson, Nov 15 2007: (Start)
The comments submitted by Miklos Kristof on Mar 19 2007 for the Fibonacci numbers (A000045) contain four important identities that have close analogs in the Lucas numbers:
For a >= b and odd b, L(a + b) + L(a - b) = 5*F(a)*F(b).
For a >= b and even b, L(a + b) + L(a - b) = L(a)*L(b).
For a >= b and odd b, L(a + b) - L(a - b) = L(a)*L(b).
For a >= b and even b, L(a + b) - L(a - b) = 5*F(a)*F(b).
A particularly interesting instance of the difference identity for even b is L(a + 30) - L(a - 30) = 5*F(a)*832040, since 5*832040 is divisible by 100, proving that the last two digits of Lucas numbers repeat in a cycle of length 60 (see A106291(100)). (End)
From John Blythe Dobson, Nov 15 2007: (Start)
The Lucas numbers satisfy remarkable difference equations, in some cases best expressed using Fibonacci numbers, of which representative examples are the following:
L(n) - L(n - 3) = 2*L(n - 2);
L(n) - L(n - 4) = 5*F(n - 2);
L(n) - L(n - 6) = 4*L(n - 3);
L(n) - L(n - 12) = 40*F(n - 6);
L(n) - L(n - 60) = 4160200*F(n - 30).
These formulas establish, respectively, that the Lucas numbers form a cyclic residue system of length 3 (mod 2), of length 4 (mod 5), of length 6 (mod 4), of length 12 (mod 40) and of length 60 (mod 4160200). The divisibility of the last modulus by 100 accounts for the fact that the last two digits of the Lucas numbers begin to repeat at L(60).
The divisibility properties of the Lucas numbers are very complex and still not fully understood, but several important criteria are established in Zhi-Hong Sun's 2003 survey of congruences for Fibonacci numbers. (End)
Sum_{n>0} a(n)/(n*2^n) = 2*log(2). - Jaume Oliver Lafont, Oct 11 2009
A010888(a(n)) = A030133(n). - Reinhard Zumkeller, Aug 20 2011
The powers of phi, the golden ratio, approach the values of the Lucas numbers, the odd powers from above and the even powers from below. - Geoffrey Caveney, Apr 18 2014
Inverse binomial transform is (-1)^n * a(n). - Michael Somos, Jun 03 2014
Lucas numbers are invariant to the following transformation for all values of the integers j and n, including negative values, thus: L(n) = (L(j+n) + (-1)^n * L(j-n))/L(j). The same transformation applied to all sequences of the form G(n+1) = m * G(n) + G(n-1) yields Lucas numbers for m = 1, except where G(j) = 0, regardless of initial values which may be nonintegers. The corresponding sequences for other values of m are: for m = 2, 2*A001333; for m = 3, A006497; for m = 4, 2*A001077; for m = 5, A087130; for m = 6, 2*A005667; for m = 7, A086902. The invariant ones all have G(0) = 2, G(1) = m. A related family of sequences is discussed at A059100. - Richard R. Forberg, Nov 23 2014
If x=a(n), y=a(n+1), z=a(n+2), then -x^2 - z*x - 3*y*x - y^2 + y*z + z^2 = 5*(-1)^(n+1). - Alexander Samokrutov, Jul 04 2015
A conjecture on the divisibility of infinite subsequences of Lucas numbers by prime(n)^m, m >= 1, is given in A266587, together with the prime "entry points". - Richard R. Forberg, Dec 31 2015
A trapezoid has three lengths of sides in order L(n-1), L(n+1), L(n-1). For increasing n a very close approximation to the maximum area will have the fourth side equal to 2*L(n). For a trapezoid with sides L(n-1), L(n-3), L(n-1), the fourth side will be L(n). - J. M. Bergot, Mar 17 2016
Satisfies Benford's law [Brown-Duncan, 1970; Berger-Hill, 2017]. - N. J. A. Sloane, Feb 08 2017
Lucas numbers L(n) and Fibonacci numbers F(n), being related by the formulas F(n) = (F(n-1) + L(n-1))/2 and L(n) = 2 F(n+1) - F(n), are a typical pair of "autosequences" (see the link to OEIS Wiki). - Jean-François Alcover, Jun 09 2017
For n >= 3, the Lucas number L(n) is the dimension of a commutative Hecke algebra of affine type A_n with independent parameters. See Theorem 1.4, Corollary 1.5, and the table on page 524 in the link "Hecke algebras with independent parameters". - Jia Huang, Jan 20 2019
From Klaus Purath, Apr 19 2019: (Start)
While all prime numbers appear as factors in the Fibonacci numbers, this is not the case with the Lucas numbers. For example, L(n) is never divisible by the following prime numbers < 150: 5, 13, 17, 37, 53, 61, 73, 89, 97, 109, 113, 137, 149 ... See A053028. Conjecture: Three properties can be determined for these prime numbers:
First observation: The prime factors > 3 occur in the Fibonacci numbers with an odd index.
Second observation: These are the prime numbers p congruent to 2, 3 (modulo 5), which occur both in Fibonacci(p+1) and in Fibonacci((p+1)/2) as prime factors, or the prime numbers p congruent to 1, 4 (modulo 5), which occur both in Fibonacci((p-1)/2) and in Fibonacci((p-1)/(2^k)) with k >= 2.
Third observation: The Pisano period lengths of these prime numbers, given in A001175, are always divisible by 4, but not by 8. In contrast, those of the prime factors of Lucas numbers are divisible either by 2, but not by 4, or by 8. (See also comment in A053028 by N. J. A. Sloane, Feb 21 2004). (End)
L(n) is the sum of 4*k consecutive terms of the Fibonacci sequence (A000045) divided by Fibonacci(2*k): (Sum_{i=0..4*k-1, k>=1} F(n+i))/F(2*k) = L(n+2*k+1). Sequences extended to negative indices, following the rule a(n-1) = a(n+1) - a(n). - Klaus Purath, Sep 15 2019
If one forms a sequence (A) of the Fibonacci type with the initial values A(0) = A022095(n) and A(1) = A000285(n), then A(n+1) = L(n+1)^2 always applies. - Klaus Purath, Sep 29 2019
From Kai Wang, Dec 18 2019: (Start)
L((2*m+1)k)/L(k) = Sum_{i=0..m-1} (-1)^(i*(k+1))*L((2*m-2*i)*k) + (-1)^(m*k).
Example: k=5, m=2, L(5)=11, L(10)=123, L(20)=15127, L(25)=167761. L(25)/L(5) = 15251, L(20) + L(10) + 1 = 15127 + 123 + 1 = 15251. (End)
From Peter Bala, Dec 23 2021: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) ( mod p^k ) hold for all prime p and positive integers n and k.
For a positive integer k, the sequence (a(n))n>=1 taken modulo k becomes a purely periodic sequence. For example, taken modulo 11, the sequence becomes [1, 3, 4, 7, 0, 7, 7, 3, 10, 2, 1, 3, 4, 7, 0, 7, 7, 3, 10, 2, ...], a periodic sequence with period 10. (End)
For any sequence with recurrence relation b(n) = b(n-1) + b(n-2), it can be shown that the recurrence relation for every k-th term is given by: b(n) = A000032(k) * b(n-k) + (-1)^(k+1) * b(n-2k), extending to negative indices as necessary. - Nick Hobson, Jan 19 2024
For n >= 3, L(n) is the number of (n-1)-digit numbers where all consecutive pairs of digits have a difference of at least 8. - Edwin Hermann, Apr 19 2025

Examples

			G.f. = 2 + x + 3*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 18*x^6 + 29*x^7 + ...

References

  • P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 69.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 32,50.
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 499.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 112, 202-203.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.5 The Fibonacci and Related Sequences, pp. 287-288.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 148.
  • Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
  • V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
  • Thomas Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, 2001.
  • C. N. Menhinick, The Fibonacci Resonance and other new Golden Ratio discoveries, Onperson, (2015), pages 200-206.
  • Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 3.
  • Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 45-46, 59.
  • Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See pp. 83-84.

Links

Crossrefs

Cf. A000204. A000045(n) = (2*L(n + 1) - L(n))/5.
First row of array A103324.
a(n) = A101220(2, 0, n), for n > 0.
a(k) = A090888(1, k) = A109754(2, k) = A118654(2, k - 1), for k > 0.
Cf. A131774, A001622, A002878 (L(2n+1)), A005248 (L(2n)), A006497, A080039, A049684 (summation of Fibonacci(4n+2)), A106291 (Pisano periods), A057854 (complement), A354265 (generalized Lucas numbers).
Cf. sequences with formula Fibonacci(n+k)+Fibonacci(n-k) listed in A280154.
Subsequence of A047201.

Programs

  • Haskell
    a000032 n = a000032_list !! n
a000032_list = 2 : 1 : zipWith (+) a000032_list (tail a000032_list)
-- Reinhard Zumkeller, Aug 20 2011
  • Magma
    [Lucas(n): n in [0..120]];
  • Maple
    with(combinat): A000032 := n->fibonacci(n+1)+fibonacci(n-1);
seq(simplify(2^n*(cos(Pi/5)^n+cos(3*Pi/5)^n)), n=0..36)
  • Mathematica
    a[0] := 2; a[n] := Nest[{Last[#], First[#] + Last[#]} &, {2, 1}, n] // Last
Array[2 Fibonacci[# + 1] - Fibonacci[#] &, 50, 0] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)
Table[LucasL[n], {n, 0, 36}] (* Zerinvary Lajos, Jul 09 2009 *)
LinearRecurrence[{1, 1}, {2, 1}, 40] (* Harvey P. Dale, Sep 07 2013 *)
LucasL[Range[0, 20]] (* Eric W. Weisstein, Aug 07 2017 *)
CoefficientList[Series[(-2 + x)/(-1 + x + x^2), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 21 2017 *)
  • PARI
    {a(n) = if(n<0, (-1)^n * a(-n), if( n<2, 2-n, a(n-1) + a(n-2)))};
  • PARI
    {a(n) = if(n<0, (-1)^n * a(-n), polsym(x^2 - x - 1, n)[n+1])};
  • PARI
    {a(n) = real((2 + quadgen(5)) * quadgen(5)^n)};
  • PARI
    a(n)=fibonacci(n+1)+fibonacci(n-1) \\ Charles R Greathouse IV, Jun 11 2011
  • PARI
    polsym(1+x-x^2, 50) \\ Charles R Greathouse IV, Jun 11 2011
  • Python
    def A000032_gen(): # generator of terms
    a, b = 2, 1
    while True:
        yield a
        a, b = b, a+b
it = A000032_gen()
A000032_list = [next(it) for  in range(50)] # _Cole Dykstra, Aug 02 2022
  • Python
    from sympy import lucas
def A000032(n): return lucas(n) # Chai Wah Wu, Sep 23 2023
  • Python
    [(i:=3)+(j:=-1)] + [(j:=i+j)+(i:=j-i) for  in range(100)] # _Jwalin Bhatt, Apr 02 2025
  • Sage
    [lucas_number2(n,1,-1) for n in range(37)] # Zerinvary Lajos, Jun 25 2008

Formula

G.f.: (2 - x)/(1 - x - x^2).
L(n) = ((1 + sqrt(5))/2)^n + ((1 - sqrt(5))/2)^n = phi^n + (1-phi)^n.
L(n) = L(n - 1) + L(n - 2) = (-1)^n * L( - n).
L(n) = Fibonacci(2*n)/Fibonacci(n) for n > 0. - Jeff Burch, Dec 11 1999
E.g.f.: 2*exp(x/2)*cosh(sqrt(5)*x/2). - Len Smiley, Nov 30 2001
L(n) = F(n) + 2*F(n - 1) = F(n + 1) + F(n - 1). - Henry Bottomley, Apr 12 2000
a(n) = sqrt(F(n)^2 + 4*F(n + 1)*F(n - 1)). - Benoit Cloitre, Jan 06 2003 [Corrected by Gary Detlefs, Jan 21 2011]
a(n) = 2^(1 - n)*Sum_{k=0..floor(n/2)} C(n, 2k)*5^k. a(n) = 2T(n, i/2)( - i)^n with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2 = - 1. - Paul Barry, Nov 15 2003
L(n) = 2*F(n + 1) - F(n). - Paul Barry, Mar 22 2004
a(n) = (phi)^n + ( - phi)^( - n). - Paul Barry, Mar 12 2005
From Miklos Kristof, Mar 19 2007: (Start)
Let F(n) = A000045 = Fibonacci numbers, L(n) = a(n) = Lucas numbers:
L(n + m) + (-1)^m*L(n - m) = L(n)*L(m).
L(n + m) - (-1)^m*L(n - m) = 8*F(n)*F(m).
L(n + m + k) + (-1)^k*L(n + m - k) + (-1)^m*(L(n - m + k) + (-1)^k*L(n - m - k)) = L(n)*L(m)*L(k).
L(n + m + k) - (-1)^k*L(n + m - k) + (-1)^m*(L(n - m + k) - (-1)^k*L(n - m - k)) = 5*F(n)*L(m)*F(k).
L(n + m + k) + (-1)^k*L(n + m - k) - (-1)^m*(L(n - m + k) + (-1)^k*L(n - m - k)) = 5*F(n)*F(m)*L(k).
L(n + m + k) - (-1)^k*L(n + m - k) - (-1)^m*(L(n - m + k) - (-1)^k*L(n - m - k)) = 5*L(n)*F(m)*F(k). (End)
Inverse: floor(log_phi(a(n)) + 1/2) = n, for n>1. Also for n >= 0, floor((1/2)*log_phi(a(n)*a(n+1))) = n. Extension valid for all integers n: floor((1/2)*sign(a(n)*a(n+1))*log_phi|a(n)*a(n+1)|) = n {where sign(x) = sign of x}. - Hieronymus Fischer, May 02 2007
Let f(n) = phi^n + phi^(-n), then L(2n) = f(2n) and L(2n + 1) = f(2n + 1) - 2*Sum_{k>=0} C(k)/f(2n + 1)^(2k + 1) where C(n) are Catalan numbers (A000108). - Gerald McGarvey, Dec 21 2007, modified by Davide Colazingari, Jul 01 2016
Starting (1, 3, 4, 7, 11, ...) = row sums of triangle A131774. - Gary W. Adamson, Jul 14 2007
a(n) = trace of the 2 X 2 matrix [0,1; 1,1]^n. - Gary W. Adamson, Mar 02 2008
From Hieronymus Fischer, Jan 02 2009: (Start)
For odd n: a(n) = floor(1/(fract(phi^n))); for even n>0: a(n) = ceiling(1/(1 - fract(phi^n))). This follows from the basic property of the golden ratio phi, which is phi - phi^(-1) = 1 (see general formula described in A001622).
a(n) = round(1/min(fract(phi^n), 1 - fract(phi^n))), for n>1, where fract(x) = x - floor(x). (End)
E.g.f.: exp(phi*x) + exp(-x/phi) with phi: = (1 + sqrt(5))/2 (golden section). 1/phi = phi - 1. See another form given in the Smiley e.g.f. comment. - Wolfdieter Lang, May 15 2010
L(n)/L(n - 1) -> A001622. - Vincenzo Librandi, Jul 17 2010
a(n) = 2*a(n-2) + a(n-3), n>2. - Gary Detlefs, Sep 09 2010
L(n) = floor(1/fract(Fibonacci(n)*phi)), for n odd. - Hieronymus Fischer, Oct 20 2010
L(n) = ceiling(1/(1 - fract(Fibonacci(n)*phi))), for n even. - Hieronymus Fischer, Oct 20 2010
L(n) = 2^n * (cos(Pi/5)^n + cos(3*Pi/5)^n). - Gary Detlefs, Nov 29 2010
L(n) = (Fibonacci(2*n - 1)*Fibonacci(2*n + 1) - 1)/(Fibonacci(n)*Fibonacci(2*n)), n != 0. - Gary Detlefs, Dec 13 2010
L(n) = sqrt(A001254(n)) = sqrt(5*Fibonacci(n)^2 - 4*(-1)^(n+1)). - Gary Detlefs, Dec 26 2010
L(n) = floor(phi^n) + ((-1)^n + 1)/2 = A014217(n) +((-1)^n+1)/2, where phi = A001622. - Gary Detlefs, Jan 20 2011
L(n) = Fibonacci(n + 6) mod Fibonacci(n + 2), n>2. - Gary Detlefs, May 19 2011
For n >= 2, a(n) = round(phi^n) where phi is the golden ratio. - Arkadiusz Wesolowski, Jul 20 2012
a(p*k) == a(k) (mod p) for primes p. a(2^s*n) == a(n)^(2^s) (mod 2) for s = 0,1,2.. a(2^k) == - 1 (mod 2^k). a(p^2*k) == a(k) (mod p) for primes p and s = 0,1,2,3.. [Hoggatt and Bicknell]. - R. J. Mathar, Jul 24 2012
From Gary Detlefs, Dec 21 2012: (Start)
L(k*n) = (F(k)*phi + F(k - 1))^n + (F(k + 1) - F(k)*phi)^n.
L(k*n) = (F(n)*phi + F(n - 1))^k + (F(n + 1) - F(n)*phi)^k.
where phi = (1 + sqrt(5))/2, F(n) = A000045(n).
(End)
L(n) = n * Sum_{k=0..floor(n/2)} binomial(n - k,k)/(n - k), n>0 [H. W. Gould]. - Gary Detlefs, Jan 20 2013
G.f.: G(0), where G(k) = 1 + 1/(1 - (x*(5*k-1))/((x*(5*k+4)) - 2/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 15 2013
L(n) = F(n) + F(n-1) + F(n-2) + F(n-3). - Bob Selcoe, Jun 17 2013
L(n) = round(sqrt(L(2n-1) + L(2n-2))). - Richard R. Forberg, Jun 24 2014
L(n) = (F(n+1)^2 - F(n-1)^2)/F(n) for n>0. - Richard R. Forberg, Nov 17 2014
L(n+2) = 1 + A001610(n+1) = 1 + Sum_{k=0..n} L(k). - Tom Edgar, Apr 15 2015
L(i+j+1) = L(i)*F(j) + L(i+1)*F(j+1) with F(n)=A000045(n). - J. M. Bergot, Feb 12 2016
a(n) = (L(n+1)^2 + 5*(-1)^n)/L(n+2). - J. M. Bergot, Apr 06 2016
Dirichlet g.f.: PolyLog(s,-1/phi) + PolyLog(s,phi), where phi is the golden ratio. - Ilya Gutkovskiy, Jul 01 2016
L(n) = F(n+2) - F(n-2). - Yuchun Ji, Feb 14 2016
L(n+1) = A087131(n+1)/2^(n+1) = 2^(-n)*Sum_{k=0..n} binomial(n,k)*5^floor((k+1)/2). - Tony Foster III, Oct 14 2017
L(2*n) = (F(k+2*n) + F(k-2*n))/F(k); n >= 1, k >= 2*n. - David James Sycamore, May 04 2018
From Greg Dresden and Shaoxiong Yuan, Jul 16 2019: (Start)
L(3n + 4)/L(3n + 1) has continued fraction: n 4's followed by a single 7.
L(3n + 3)/L(3n) has continued fraction: n 4's followed by a single 2.
L(3n + 2)/L(3n - 1) has continued fraction: n 4's followed by a single -3. (End)
From Klaus Purath, Sep 15 2019: (Start)
All involved sequences extended to negative indices, following the rule a(n-1) = a(n+1) - a(n).
L(n) = (2*L(n+2) - L(n-3))/5.
L(n) = (2*L(n-2) + L(n+3))/5.
L(n) = F(n-3) + 2*F(n).
L(n) = 2*F(n+2) - 3*F(n).
L(n) = (3*F(n-1) + F(n+2))/2.
L(n) = 3*F(n-3) + 4*F(n-2).
L(n) = 4*F(n+1) - F(n+3).
L(n) = (F(n-k) + F(n+k))/F(k) with odd k>0.
L(n) = (F(n+k) - F(n-k))/F(k) with even k>0.
L(n) = A001060(n-1) - F(n+1).
L(n) = (A022121(n-1) - F(n+1))/2.
L(n) = (A022131(n-1) - F(n+1))/3.
L(n) = (A022139(n-1) - F(n+1))/4.
L(n) = (A166025(n-1) - F(n+1))/5.
The following two formulas apply for all sequences of the Fibonacci type.
(a(n-2*k) + a(n+2*k))/a(n) = L(2*k).
(a(n+2*k+1) - a(n-2*k-1))/a(n) = L(2*k+1). (End)
L(n) = F(n-k)*L(k+1) + F(n-k-1)*L(k), for all k >= 0, where F(n) = A000045(n). - Michael Tulskikh, Dec 06 2019
F(n+2*m) = L(m)*F(n+m) + (-1)^(m-1)*F(n) for all n >= 0 and m >= 0. - Alexander Burstein, Mar 31 2022
a(n) = i^(n-1)*cos(n*c)/cos(c) = i^(n-1)*cos(c*n)*sec(c), where c = Pi/2 + i*arccsch(2). - Peter Luschny, May 23 2022
From Yike Li and Greg Dresden, Aug 25 2022: (Start)
L(2*n) = 5*binomial(2*n-1,n) - 2^(2*n-1) + 5*Sum_{j=1..n/5} binomial(2*n,n+5*j) for n>0.
L(2*n+1) = 2^(2n) - 5*Sum_{j=0..n/5} binomial(2*n+1,n+5*j+3). (End)
From Andrea Pinos, Jul 04 2023: (Start)
L(n) ~ Gamma(1/phi^n) + gamma.
L(n) = Re(phi^n + e^(i*Pi*n)/phi^n). (End)
L(n) = ((Sum_{i=0..n-1} L(i)^2) - 2)/L(n-1). - Jules Beauchamp, May 03 2025
From Peter Bala, Jul 09 2025: (Start)
The following series telescope:
For k >= 1, Sum_{n >= 1} (-1)^((k+1)*(n+1)) * a(2*n*k)/(a((2*n-1)*k)*a((2*n+1)*k)) = 1/a(k)^2.
For positive even k, Sum_{n >= 1} 1/(a(k*n) - (a(k) + 2)/a(k*n)) = 1/(a(k) - 2) and
Sum_{n >= 1} (-1)^(n+1)/(a(k*n) + (a(k) - 2)/a(k*n)) = 1/(a(k) + 2).
For positive odd k, Sum_{n >= 1} 1/(a(k*n) - (-1)^n*(a(2*k) + 2)/a(k*n)) = (a(k) + 2)/(2*(a(2*k) - 2)) and
Sum_{n >= 1} (-1)^(n+1)/(a(k*n) - (-1)^n*(a(2*k) + 2)/a(k*n)) = (a(k) - 2)/(2*(a(2*k) - 2)). (End)

A001147 Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1).

Original entry on oeis.org

1, 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075, 13749310575, 316234143225, 7905853580625, 213458046676875, 6190283353629375, 191898783962510625, 6332659870762850625, 221643095476699771875, 8200794532637891559375, 319830986772877770815625
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The solution to Schröder's third problem.
Number of fixed-point-free involutions in symmetric group S_{2n} (cf. A000085).
a(n-2) is the number of full Steiner topologies on n points with n-2 Steiner points. [corrected by Lyle Ramshaw, Jul 20 2022]
a(n) is also the number of perfect matchings in the complete graph K(2n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 25 2001
Number of ways to choose n disjoint pairs of items from 2*n items. - Ron Zeno (rzeno(AT)hotmail.com), Feb 06 2002
Number of ways to choose n-1 disjoint pairs of items from 2*n-1 items (one item remains unpaired). - Bartosz Zoltak, Oct 16 2012
For n >= 1 a(n) is the number of permutations in the symmetric group S_(2n) whose cycle decomposition is a product of n disjoint transpositions. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 21 2001
a(n) is the number of distinct products of n+1 variables with commutative, nonassociative multiplication. - Andrew Walters (awalters3(AT)yahoo.com), Jan 17 2004. For example, a(3)=15 because the product of the four variables w, x, y and z can be constructed in exactly 15 ways, assuming commutativity but not associativity: 1. w(x(yz)) 2. w(y(xz)) 3. w(z(xy)) 4. x(w(yz)) 5. x(y(wz)) 6. x(z(wy)) 7. y(w(xz)) 8. y(x(wz)) 9. y(z(wx)) 10. z(w(xy)) 11. z(x(wy)) 12. z(y(wx)) 13. (wx)(yz) 14. (wy)(xz) 15. (wz)(xy).
a(n) = E(X^(2n)), where X is a standard normal random variable (i.e., X is normal with mean = 0, variance = 1). So for instance a(3) = E(X^6) = 15, etc. See Abramowitz and Stegun or Hoel, Port and Stone. - Jerome Coleman, Apr 06 2004
Second Eulerian transform of 1,1,1,1,1,1,... The second Eulerian transform transforms a sequence s to a sequence t by the formula t(n) = Sum_{k=0..n} E(n,k)s(k), where E(n,k) is a second-order Eulerian number (A008517). - Ross La Haye, Feb 13 2005
Integral representation as n-th moment of a positive function on the positive axis: a(n) = Integral_{x=0..oo} x^n*exp(-x/2)/sqrt(2*Pi*x) dx, n >= 0. - Karol A. Penson, Oct 10 2005
a(n) is the number of binary total partitions of n+1 (each non-singleton block must be partitioned into exactly two blocks) or, equivalently, the number of unordered full binary trees with n+1 labeled leaves (Stanley, ex 5.2.6). - Mitch Harris, Aug 01 2006
a(n) is the Pfaffian of the skew-symmetric 2n X 2n matrix whose (i,j) entry is i for iDavid Callan, Sep 25 2006
a(n) is the number of increasing ordered rooted trees on n+1 vertices where "increasing" means the vertices are labeled 0,1,2,...,n so that each path from the root has increasing labels. Increasing unordered rooted trees are counted by the factorial numbers A000142. - David Callan, Oct 26 2006
Number of perfect multi Skolem-type sequences of order n. - Emeric Deutsch, Nov 24 2006
a(n) = total weight of all Dyck n-paths (A000108) when each path is weighted with the product of the heights of the terminal points of its upsteps. For example with n=3, the 5 Dyck 3-paths UUUDDD, UUDUDD, UUDDUD, UDUUDD, UDUDUD have weights 1*2*3=6, 1*2*2=4, 1*2*1=2, 1*1*2=2, 1*1*1=1 respectively and 6+4+2+2+1=15. Counting weights by height of last upstep yields A102625. - David Callan, Dec 29 2006
a(n) is the number of increasing ternary trees on n vertices. Increasing binary trees are counted by ordinary factorials (A000142) and increasing quaternary trees by triple factorials (A007559). - David Callan, Mar 30 2007
From Tom Copeland, Nov 13 2007, clarified in first and extended in second paragraph, Jun 12 2021: (Start)
a(n) has the e.g.f. (1-2x)^(-1/2) = 1 + x + 3*x^2/2! + ..., whose reciprocal is (1-2x)^(1/2) = 1 - x - x^2/2! - 3*x^3/3! - ... = b(0) - b(1)*x - b(2)*x^2/2! - ... with b(0) = 1 and b(n+1) = -a(n) otherwise. By the formalism of A133314, Sum_{k=0..n} binomial(n,k)*b(k)*a(n-k) = 0^n where 0^0 := 1. In this sense, the sequence a(n) is essentially self-inverse. See A132382 for an extension of this result. See A094638 for interpretations.
This sequence aerated has the e.g.f. e^(t^2/2) = 1 + t^2/2! + 3*t^4/4! + ... = c(0) + c(1)*t + c(2)*t^2/2! + ... and the reciprocal e^(-t^2/2); therefore, Sum_{k=0..n} cos(Pi k/2)*binomial(n,k)*c(k)*c(n-k) = 0^n; i.e., the aerated sequence is essentially self-inverse. Consequently, Sum_{k=0..n} (-1)^k*binomial(2n,2k)*a(k)*a(n-k) = 0^n. (End)
From Ross Drewe, Mar 16 2008: (Start)
This is also the number of ways of arranging the elements of n distinct pairs, assuming the order of elements is significant but the pairs are not distinguishable, i.e., arrangements which are the same after permutations of the labels are equivalent.
If this sequence and A000680 are denoted by a(n) and b(n) respectively, then a(n) = b(n)/n! where n! = the number of ways of permuting the pair labels.
For example, there are 90 ways of arranging the elements of 3 pairs [1 1], [2 2], [3 3] when the pairs are distinguishable: A = { [112233], [112323], ..., [332211] }.
By applying the 6 relabeling permutations to A, we can partition A into 90/6 = 15 subsets: B = { {[112233], [113322], [221133], [223311], [331122], [332211]}, {[112323], [113232], [221313], [223131], [331212], [332121]}, ....}
Each subset or equivalence class in B represents a unique pattern of pair relationships. For example, subset B1 above represents {3 disjoint pairs} and subset B2 represents {1 disjoint pair + 2 interleaved pairs}, with the order being significant (contrast A132101). (End)
A139541(n) = a(n) * a(2*n). - Reinhard Zumkeller, Apr 25 2008
a(n+1) = Sum_{j=0..n} A074060(n,j) * 2^j. - Tom Copeland, Sep 01 2008
From Emeric Deutsch, Jun 05 2009: (Start)
a(n) is the number of adjacent transpositions in all fixed-point-free involutions of {1,2,...,2n}. Example: a(2)=3 because in 2143=(12)(34), 3412=(13)(24), and 4321=(14)(23) we have 2 + 0 + 1 adjacent transpositions.
a(n) = Sum_{k>=0} k*A079267(n,k).
(End)
Hankel transform is A137592. - Paul Barry, Sep 18 2009
(1, 3, 15, 105, ...) = INVERT transform of A000698 starting (1, 2, 10, 74, ...). - Gary W. Adamson, Oct 21 2009
a(n) = (-1)^(n+1)*H(2*n,0), where H(n,x) is the probabilists' Hermite polynomial. The generating function for the probabilists' Hermite polynomials is as follows: exp(x*t-t^2/2) = Sum_{i>=0} H(i,x)*t^i/i!. - Leonid Bedratyuk, Oct 31 2009
The Hankel transform of a(n+1) is A168467. - Paul Barry, Dec 04 2009
Partial products of odd numbers. - Juri-Stepan Gerasimov, Oct 17 2010
See A094638 for connections to differential operators. - Tom Copeland, Sep 20 2011
a(n) is the number of subsets of {1,...,n^2} that contain exactly k elements from {1,...,k^2} for k=1,...,n. For example, a(3)=15 since there are 15 subsets of {1,2,...,9} that satisfy the conditions, namely, {1,2,5}, {1,2,6}, {1,2,7}, {1,2,8}, {1,2,9}, {1,3,5}, {1,3,6}, {1,3,7}, {1,3,8}, {1,3,9}, {1,4,5}, {1,4,6}, {1,4,7}, {1,4,8}, and {1,4,9}. - Dennis P. Walsh, Dec 02 2011
a(n) is the leading coefficient of the Bessel polynomial y_n(x) (cf. A001498). - Leonid Bedratyuk, Jun 01 2012
For n>0: a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = min(i,j)^2 for 1 <= i,j <= n. - Enrique Pérez Herrero, Jan 14 2013
a(n) is also the numerator of the mean value from 0 to Pi/2 of sin(x)^(2n). - Jean-François Alcover, Jun 13 2013
a(n) is the size of the Brauer monoid on 2n points (see A227545). - James Mitchell, Jul 28 2013
For n>1: a(n) is the numerator of M(n)/M(1) where the numbers M(i) have the property that M(n+1)/M(n) ~ n-1/2 (for example, large Kendell-Mann numbers, see A000140 or A181609, as n --> infinity). - Mikhail Gaichenkov, Jan 14 2014
a(n) = the number of upper-triangular matrix representations required for the symbolic representation of a first order central moment of the multivariate normal distribution of dimension 2(n-1), i.e., E[X_1*X_2...*X_(2n-2)|mu=0, Sigma]. See vignette for symmoments R package on CRAN and Phillips reference below. - Kem Phillips, Aug 10 2014
For n>1: a(n) is the number of Feynman diagrams of order 2n (number of internal vertices) for the vacuum polarization with one charged loop only, in quantum electrodynamics. - Robert Coquereaux, Sep 15 2014
Aerated with intervening zeros (1,0,1,0,3,...) = a(n) (cf. A123023), the e.g.f. is e^(t^2/2), so this is the base for the Appell sequence A099174 with e.g.f. e^(t^2/2) e^(x*t) = exp(P(.,x),t) = unsigned A066325(x,t), the probabilist's (or normalized) Hermite polynomials. P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for A066325(x,t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), where UP(n,t) are the polynomials of A066325 and, e.g., (P(.,t))^n = P(n,t). - Tom Copeland, Nov 15 2014
a(n) = the number of relaxed compacted binary trees of right height at most one of size n. A relaxed compacted binary tree of size n is a directed acyclic graph consisting of a binary tree with n internal nodes, one leaf, and n pointers. It is constructed from a binary tree of size n, where the first leaf in a post-order traversal is kept and all other leaves are replaced by pointers. These links may point to any node that has already been visited by the post-order traversal. The right height is the maximal number of right-edges (or right children) on all paths from the root to any leaf after deleting all pointers. The number of unbounded relaxed compacted binary trees of size n is A082161(n). See the Genitrini et al. link. - Michael Wallner, Jun 20 2017
Also the number of distinct adjacency matrices in the n-ladder rung graph. - Eric W. Weisstein, Jul 22 2017
From Christopher J. Smyth, Jan 26 2018: (Start)
a(n) = the number of essentially different ways of writing a probability distribution taking n+1 values as a sum of products of binary probability distributions. See comment of Mitch Harris above. This is because each such way corresponds to a full binary tree with n+1 leaves, with the leaves labeled by the values. (This comment is due to Niko Brummer.)
Also the number of binary trees with root labeled by an (n+1)-set S, its n+1 leaves by the singleton subsets of S, and other nodes labeled by subsets T of S so that the two daughter nodes of the node labeled by T are labeled by the two parts of a 2-partition of T. This also follows from Mitch Harris' comment above, since the leaf labels determine the labels of the other vertices of the tree.
(End)
a(n) is the n-th moment of the chi-squared distribution with one degree of freedom (equivalent to Coleman's Apr 06 2004 comment). - Bryan R. Gillespie, Mar 07 2021
Let b(n) = 0 for n odd and b(2k) = a(k); i.e., let the sequence b(n) be an aerated version of this entry. After expanding the differential operator (x + D)^n and normal ordering the resulting terms, the integer coefficient of the term x^k D^m is n! b(n-k-m) / [(n-k-m)! k! m!] with 0 <= k,m <= n and (k+m) <= n. E.g., (x+D)^2 = x^2 + 2xD + D^2 + 1 with D = d/dx. The result generalizes to the raising (R) and lowering (L) operators of any Sheffer polynomial sequence by replacing x by R and D by L and follows from the disentangling relation e^{t(L+R)} = e^{t^2/2} e^{tR} e^{tL}. Consequently, these are also the coefficients of the reordered 2^n permutations of the binary symbols L and R under the condition LR = RL + 1. E.g., (L+R)^2 = LL + LR + RL + RR = LL + 2RL + RR + 1. (Cf. A344678.) - Tom Copeland, May 25 2021
From Tom Copeland, Jun 14 2021: (Start)
Lando and Zvonkin present several scenarios in which the double factorials occur in their role of enumerating perfect matchings (pairings) and as the nonzero moments of the Gaussian e^(x^2/2).
Speyer and Sturmfels (p. 6) state that the number of facets of the abstract simplicial complex known as the tropical Grassmannian G'''(2,n), the space of phylogenetic T_n trees (see A134991), or Whitehouse complex is a shifted double factorial.
These are also the unsigned coefficients of the x[2]^m terms in the partition polynomials of A134685 for compositional inversion of e.g.f.s, a refinement of A134991.
a(n)*2^n = A001813(n) and A001813(n)/(n+1)! = A000108(n), the Catalan numbers, the unsigned coefficients of the x[2]^m terms in the partition polynomials A133437 for compositional inversion of o.g.f.s, a refinement of A033282, A126216, and A086810. Then the double factorials inherit a multitude of analytic and combinatoric interpretations from those of the Catalan numbers, associahedra, and the noncrossing partitions of A134264 with the Catalan numbers as unsigned-row sums. (End)
Connections among the Catalan numbers A000108, the odd double factorials, values of the Riemann zeta function and its derivative for integer arguments, and series expansions of the reduced action for the simple harmonic oscillator and the arc length of the spiral of Archimedes are given in the MathOverflow post on the Riemann zeta function. - Tom Copeland, Oct 02 2021
b(n) = a(n) / (n! 2^n) = Sum_{k = 0..n} (-1)^n binomial(n,k) (-1)^k a(k) / (k! 2^k) = (1-b.)^n, umbrally; i.e., the normalized double factorial a(n) is self-inverse under the binomial transform. This can be proved by applying the Euler binomial transformation for o.g.f.s Sum_{n >= 0} (1-b.)^n x^n = (1/(1-x)) Sum_{n >= 0} b_n (x / (x-1))^n to the o.g.f. (1-x)^{-1/2} = Sum_{n >= 0} b_n x^n. Other proofs are suggested by the discussion in Watson on pages 104-5 of transformations of the Bessel functions of the first kind with b(n) = (-1)^n binomial(-1/2,n) = binomial(n-1/2,n) = (2n)! / (n! 2^n)^2. - Tom Copeland, Dec 10 2022

Examples

			a(3) = 1*3*5 = 15.
From _Joerg Arndt_, Sep 10 2013: (Start)
There are a(3)=15 involutions of 6 elements without fixed points:
  #:    permutation           transpositions
  01:  [ 1 0 3 2 5 4 ]      (0, 1) (2, 3) (4, 5)
  02:  [ 1 0 4 5 2 3 ]      (0, 1) (2, 4) (3, 5)
  03:  [ 1 0 5 4 3 2 ]      (0, 1) (2, 5) (3, 4)
  04:  [ 2 3 0 1 5 4 ]      (0, 2) (1, 3) (4, 5)
  05:  [ 2 4 0 5 1 3 ]      (0, 2) (1, 4) (3, 5)
  06:  [ 2 5 0 4 3 1 ]      (0, 2) (1, 5) (3, 4)
  07:  [ 3 2 1 0 5 4 ]      (0, 3) (1, 2) (4, 5)
  08:  [ 3 4 5 0 1 2 ]      (0, 3) (1, 4) (2, 5)
  09:  [ 3 5 4 0 2 1 ]      (0, 3) (1, 5) (2, 4)
  10:  [ 4 2 1 5 0 3 ]      (0, 4) (1, 2) (3, 5)
  11:  [ 4 3 5 1 0 2 ]      (0, 4) (1, 3) (2, 5)
  12:  [ 4 5 3 2 0 1 ]      (0, 4) (1, 5) (2, 3)
  13:  [ 5 2 1 4 3 0 ]      (0, 5) (1, 2) (3, 4)
  14:  [ 5 3 4 1 2 0 ]      (0, 5) (1, 3) (2, 4)
  15:  [ 5 4 3 2 1 0 ]      (0, 5) (1, 4) (2, 3)
(End)
G.f. = 1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + 135135*x^7 + ...

References

  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, (26.2.28).
  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 317.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 228, #19.
  • Hoel, Port and Stone, Introduction to Probability Theory, Section 7.3.
  • F. K. Hwang, D. S. Richards and P. Winter, The Steiner Tree Problem, North-Holland, 1992, see p. 14.
  • C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980, pages 466-467.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.6 and also p. 178.
  • R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer-Verlag, New York, 1999, p. 73.
  • G. Watson, The Theory of Bessel Functions, Cambridge Univ. Press, 1922.

Links

Crossrefs

Cf. A000085, A006882, A000165 ((2n)!!), A001818, A009445, A039683, A102992, A001190 (no labels), A000680, A132101.
Cf. A086677; A055142 (for this sequence, |a(n+1)| + 1 is the number of distinct products which can be formed using commutative, nonassociative multiplication and a nonempty subset of n given variables).
Constant terms of polynomials in A098503. First row of array A099020.
Cf. A079267, A000698, A029635, A161198, A076795, A123023, A161124, A051125, A181983, A099174, A087547, A028338 (first column).
Subsequence of A248652.
Cf. A082161 (relaxed compacted binary trees of unbounded right height).
Cf. A053871 (binomial transform).
Cf. A000108, A001813, A033282, A060540, A086810, A094638, A126216, A133437, A134264, A134685, A134991, A344678.

Programs

  • GAP
    A001147 := function(n) local i, s, t; t := 1; i := 0; Print(t, ", "); for i in [1 .. n] do t := t*(2*i-1); Print(t, ", "); od; end; A001147(100); # Stefano Spezia, Nov 13 2018
  • Haskell
    a001147 n = product [1, 3 .. 2 * n - 1]
a001147_list = 1 : zipWith (*) [1, 3 ..] a001147_list
-- Reinhard Zumkeller, Feb 15 2015, Dec 03 2011
  • Magma
    A001147:=func< n | n eq 0 select 1 else &*[ k: k in [1..2*n-1 by 2] ] >; [ A001147(n): n in [0..20] ]; // Klaus Brockhaus, Jun 22 2011
  • Magma
    I:=[1,3]; [1] cat [n le 2 select I[n]  else (3*n-2)*Self(n-1)-(n-1)*(2*n-3)*Self(n-2): n in [1..25] ]; // Vincenzo Librandi, Feb 19 2015
  • Maple
    f := n->(2*n)!/(n!*2^n);
A001147 := proc(n) doublefactorial(2*n-1); end: # R. J. Mathar, Jul 04 2009
A001147 := n -> 2^n*pochhammer(1/2, n); # Peter Luschny, Aug 09 2009
G(x):=(1-2*x)^(-1/2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # Zerinvary Lajos, Apr 03 2009; aligned with offset by Johannes W. Meijer, Aug 11 2009
series(hypergeom([1,1/2],[],2*x),x=0,20); # Mark van Hoeij, Apr 07 2013
  • Mathematica
    Table[(2 n - 1)!!, {n, 0, 19}] (* Robert G. Wilson v, Oct 12 2005 *)
a[ n_] := 2^n Gamma[n + 1/2] / Gamma[1/2]; (* Michael Somos, Sep 18 2014 *)
Join[{1}, Range[1, 41, 2]!!] (* Harvey P. Dale, Jan 28 2017 *)
a[ n_] := If[ n < 0, (-1)^n / a[-n], SeriesCoefficient[ Product[1 - (1 - x)^(2 k - 1), {k, n}], {x, 0, n}]]; (* Michael Somos, Jun 27 2017 *)
(2 Range[0, 20] - 1)!! (* Eric W. Weisstein, Jul 22 2017 *)
  • Maxima
    a(n):=if n=0 then 1 else sum(sum(binomial(n-1,i)*binomial(n-i-1,j)*a(i)*a(j)*a(n-i-j-1),j,0,n-i-1),i,0,n-1); /* Vladimir Kruchinin, May 06 2020 */
  • PARI
    {a(n) = if( n<0, (-1)^n / a(-n), (2*n)! / n! / 2^n)}; /* Michael Somos, Sep 18 2014 */
  • PARI
    x='x+O('x^33); Vec(serlaplace((1-2*x)^(-1/2))) \\ Joerg Arndt, Apr 24 2011
  • Python
    from sympy import factorial2
def a(n): return factorial2(2 * n - 1)
print([a(n) for n in range(101)])  # Indranil Ghosh, Jul 22 2017
  • Sage
    [rising_factorial(n+1,n)/2^n for n in (0..15)] # Peter Luschny, Jun 26 2012

Formula

E.g.f.: 1 / sqrt(1 - 2*x).
D-finite with recurrence: a(n) = a(n-1)*(2*n-1) = (2*n)!/(n!*2^n) = A010050(n)/A000165(n).
a(n) ~ sqrt(2) * 2^n * (n/e)^n.
Rational part of numerator of Gamma(n+1/2): a(n) * sqrt(Pi) / 2^n = Gamma(n+1/2). - Yuriy Brun, Ewa Dominowska (brun(AT)mit.edu), May 12 2001
With interpolated zeros, the sequence has e.g.f. exp(x^2/2). - Paul Barry, Jun 27 2003
The Ramanujan polynomial psi(n+1, n) has value a(n). - Ralf Stephan, Apr 16 2004
a(n) = Sum_{k=0..n} (-2)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005
Log(1 + x + 3*x^2 + 15*x^3 + 105*x^4 + 945*x^5 + 10395*x^6 + ...) = x + 5/2*x^2 + 37/3*x^3 + 353/4*x^4 + 4081/5*x^5 + 55205/6*x^6 + ..., where [1, 5, 37, 353, 4081, 55205, ...] = A004208. - Philippe Deléham, Jun 20 2006
1/3 + 2/15 + 3/105 + ... = 1/2. [Jolley eq. 216]
Sum_{j=1..n} j/a(j+1) = (1 - 1/a(n+1))/2. [Jolley eq. 216]
1/1 + 1/3 + 2/15 + 6/105 + 24/945 + ... = Pi/2. - Gary W. Adamson, Dec 21 2006
a(n) = (1/sqrt(2*Pi))*Integral_{x>=0} x^n*exp(-x/2)/sqrt(x). - Paul Barry, Jan 28 2008
a(n) = A006882(2n-1). - R. J. Mathar, Jul 04 2009
G.f.: 1/(1-x-2x^2/(1-5x-12x^2/(1-9x-30x^2/(1-13x-56x^2/(1- ... (continued fraction). - Paul Barry, Sep 18 2009
a(n) = (-1)^n*subs({log(e)=1,x=0},coeff(simplify(series(e^(x*t-t^2/2),t,2*n+1)),t^(2*n))*(2*n)!). - Leonid Bedratyuk, Oct 31 2009
a(n) = 2^n*gamma(n+1/2)/gamma(1/2). - Jaume Oliver Lafont, Nov 09 2009
G.f.: 1/(1-x/(1-2x/(1-3x/(1-4x/(1-5x/(1- ...(continued fraction). - Aoife Hennessy (aoife.hennessy(AT)gmail.com), Dec 02 2009
The g.f. of a(n+1) is 1/(1-3x/(1-2x/(1-5x/(1-4x/(1-7x/(1-6x/(1-.... (continued fraction). - Paul Barry, Dec 04 2009
a(n) = Sum_{i=1..n} binomial(n,i)*a(i-1)*a(n-i). - Vladimir Shevelev, Sep 30 2010
E.g.f.: A(x) = 1 - sqrt(1-2*x) satisfies the differential equation A'(x) - A'(x)*A(x) - 1 = 0. - Vladimir Kruchinin, Jan 17 2011
a(n) = A123023(2*n). - Michael Somos, Jul 24 2011
a(n) = (1/2)*Sum_{i=1..n} binomial(n+1,i)*a(i-1)*a(n-i). See link above. - Dennis P. Walsh, Dec 02 2011
a(n) = Sum_{k=0..n} (-1)^k*binomial(2*n,n+k)*Stirling_1(n+k,k) [Kauers and Ko].
a(n) = A035342(n, 1), n >= 1 (first column of triangle).
a(n) = A001497(n, 0) = A001498(n, n), first column, resp. main diagonal, of Bessel triangle.
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^n and sum of top row terms of M^(n-1), where M = a variant of the (1,2) Pascal triangle (Cf. A029635) as the following production matrix:
1, 2, 0, 0, 0, ...
1, 3, 2, 0, 0, ...
1, 4, 5, 2, 0, ...
1, 5, 9, 7, 2, ...
...
For example, a(3) = 15 is the left term in top row of M^3: (15, 46, 36, 8) and a(4) = 105 = (15 + 46 + 36 + 8).
(End)
G.f.: A(x) = 1 + x/(W(0) - x); W(k) = 1 + x + x*2*k - x*(2*k + 3)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2011
a(n) = Sum_{i=1..n} binomial(n,i-1)*a(i-1)*a(n-i). - Dennis P. Walsh, Dec 02 2011
a(n) = A009445(n) / A014481(n). - Reinhard Zumkeller, Dec 03 2011
a(n) = (-1)^n*Sum_{k=0..n} 2^(n-k)*s(n+1,k+1), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
a(n) = (2*n)4! = Gauss_factorial(2*n,4) = Product{j=1..2*n, gcd(j,4)=1} j. - Peter Luschny, Oct 01 2012
G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(2*k - 1)/(1 - x*(2*k + 2)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: 1 + x/Q(0), where Q(k) = 1 + (2*k - 1)*x - 2*x*(k + 1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 2*x*(2*k + 1)/(2*x*(2*k + 1) - 1 + 2*x*(2*k + 2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x/(x + 1/(2*k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: G(0), where G(k) = 1 + 2*x*(4*k + 1)/(4*k + 2 - 2*x*(2*k + 1)*(4*k + 3)/(x*(4*k + 3) + 2*(k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 22 2013
a(n) = (2*n - 3)*a(n-2) + (2*n - 2)*a(n-1), n > 1. - Ivan N. Ianakiev, Jul 08 2013
G.f.: G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 04 2013
a(n) = 2*a(n-1) + (2n-3)^2*a(n-2), a(0) = a(1) = 1. - Philippe Deléham, Oct 27 2013
G.f. of reciprocals: Sum_{n>=0} x^n/a(n) = 1F1(1; 1/2; x/2), confluent hypergeometric Function. - R. J. Mathar, Jul 25 2014
0 = a(n)*(+2*a(n+1) - a(n+2)) + a(n+1)*(+a(n+1)) for all n in Z. - Michael Somos, Sep 18 2014
a(n) = (-1)^n / a(-n) = 2*a(n-1) + a(n-1)^2 / a(n-2) for all n in Z. - Michael Somos, Sep 18 2014
From Peter Bala, Feb 18 2015: (Start)
Recurrence equation: a(n) = (3*n - 2)*a(n-1) - (n - 1)*(2*n - 3)*a(n-2) with a(1) = 1 and a(2) = 3.
The sequence b(n) = A087547(n), beginning [1, 4, 52, 608, 12624, ... ], satisfies the same second-order recurrence equation. This leads to the generalized continued fraction expansion lim_{n -> infinity} b(n)/a(n) = Pi/2 = 1 + 1/(3 - 6/(7 - 15/(10 - ... - n*(2*n - 1)/((3*n + 1) - ... )))). (End)
E.g.f of the sequence whose n-th element (n = 1,2,...) equals a(n-1) is 1-sqrt(1-2*x). - Stanislav Sykora, Jan 06 2017
Sum_{n >= 1} a(n)/(2*n-1)! = exp(1/2). - Daniel Suteu, Feb 06 2017
a(n) = A028338(n, 0), n >= 0. - Wolfdieter Lang, May 27 2017
a(n) = (Product_{k=0..n-2} binomial(2*(n-k),2))/n!. - Stefano Spezia, Nov 13 2018
a(n) = Sum_{i=0..n-1} Sum_{j=0..n-i-1} C(n-1,i)*C(n-i-1,j)*a(i)*a(j)*a(n-i-j-1), a(0)=1, - Vladimir Kruchinin, May 06 2020
From Amiram Eldar, Jun 29 2020: (Start)
Sum_{n>=1} 1/a(n) = sqrt(e*Pi/2)*erf(1/sqrt(2)), where erf is the error function.
Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(Pi/(2*e))*erfi(1/sqrt(2)), where erfi is the imaginary error function. (End)
G.f. of reciprocals: R(x) = Sum_{n>=0} x^n/a(n) satisfies (1 + x)*R(x) = 1 + 2*x*R'(x). - Werner Schulte, Nov 04 2024

Extensions

Removed erroneous comments: neither the number of n X n binary matrices A such that A^2 = 0 nor the number of simple directed graphs on n vertices with no directed path of length two are counted by this sequence (for n = 3, both are 13). - Dan Drake, Jun 02 2009

A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 9, 5, 1, 1, 9, 16, 14, 6, 1, 1, 11, 25, 30, 20, 7, 1, 1, 13, 36, 55, 50, 27, 8, 1, 1, 15, 49, 91, 105, 77, 35, 9, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1
Offset: 1

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Author

Clark Kimberling, Feb 28 2012

Keywords

Comments

Row sums: A083329
Alternating row sums: 1,0,-1,-1,-1,-1,-1,-1,-1,-1,...
Antidiagonal sums: A000071 (-1+Fibonacci numbers)
col 1: A000012
col 2: A005408
col 3: A000290
col 4: A000330
col 5: A002415
col 6: A005585
col 7: A040977
col 8: A050486
col 9: A053347
col 10: A054333
col 11: A054334
col 12: A057788
col 2n-1 of A208510 is column n of A208508
col 2n of A208510 is column n of A208509.
...
GENERAL DISCUSSION:
A208510 typifies arrays generated by paired recurrence equations of the following form:
u(n,x)=a(n,x)*u(n-1,x)+b(n,x)*v(n-1,x)+c(n,x)
v(n,x)=d(n,x)*u(n-1,x)+e(n,x)*v(n-1,x)+f(n,x).
...
These first-order recurrences imply separate second-order recurrences. In order to show them, the six functions a(n,x),...,f(n,x) are abbreviated as a,b,c,d,e,f.
Then, starting with initial values u(1,x)=1 and u(2,x)=a+b+c: u(n,x) = (a+e)u(n-1,x) + (bd-ae)u(n-2,x) + bf-ce+c.
With initial values v(1,x)=1 and v(2,x)=d+e+f: v(n,x) = (a+e)v(n-1,x) + (bd-ae)v(n-2,x) + cd-af+f.
...
In the guide below, the last column codes certain sequences that occur in one of these ways: row, column, edge, row sum, alternating row sum. Coding:
A: 1,-1,1,-1,1,-1,1.... A033999
B: 1,2,4,8,16,32,64,... powers of 2
C: 1,1,1,1,1,1,1,1,.... A000012
D: 2,2,2,2,2,2,2,2,.... A007395
E: 2,4,6,8,10,12,14,... even numbers
F: 1,1,2,3,5,8,13,21,.. Fibonacci numbers
N: 1,2,3,4,5,6,7,8,.... A000027
O: 1,3,5,7,9,11,13,.... odd numbers
P: 1,3,9,27,81,243,.... powers of 3
S: 1,4,9,16,25,36,49,.. squares
T: 1,3,6,10,15,21,38,.. triangular numbers
Z: 1,0,0,0,0,0,0,0,0,.. A000007
*: (eventually) periodic alternating row sums
^: has a limiting row; i.e., the polynomials "approach" a power series
This coding includes indirect and repeated occurrences; e.g. F occurs thrice at A094441: in column 1 directly as Fibonacci numbers, in row sums as odd-indexed Fibonacci numbers, and in alternating row sums as signed Fibonacci numbers.
......... a....b....c....d....e....f....code
A034839 u 1....1....0....1....x....0....CCOT
A034867 v 1....1....0....1....x....0....CEN
A210221 u 1....1....0....1....2x...0....BBFF
A210596 v 1....1....0....1....2x...0....BBFF
A105070 v 1....2x...0....1....1....0....BN
A207605 u 1....1....0....1....x+1..0....BCFFN
A106195 v 1....1....0....1....x+1..0....BCFFN
A207606 u 1....1....0....x....x+1..0....DNT
A207607 v 1....1....0....x....x+1..0....DNT
A207608 u 1....1....0....2x...x+1..0....N
A207609 v 1....1....0....2x...x+1..0....C
A207610 u 1....1....0....1....x....1....CF
A207611 v 1....1....0....1....x....1....BCF
A207612 u 1....1....0....1....2x...1....BF
A207613 v 1....1....0....1....2x...1....BF
A207614 u 1....1....0....1....x+1..1....CN
A207615 v 1....1....0....1....x+1..1....CFN
A207616 u 1....1....0....x....1....1....CE
A207617 v 1....1....0....x....1....1....CNO
A029638 u 1....1....0....x....x....1....CDNO
A029635 v 1....1....0....x....x....1....CDNOZ
A207618 u 1....1....0....x....2x...1....N
A207619 v 1....1....0....x....2x...1....CFN
A207620 u 1....1....0....x....x+1..1....DET
A207621 v 1....1....0....x....x+1..1....DNO
A207622 u 1....1....0....2x...1....1....BT
A207623 v 1....1....0....2x...1....1....BN
A207624 u 1....1....0....2x...x....1....N
A102662 v 1....1....0....2x...x....1....CO
A207625 u 1....1....0....2x...x+1..1....T
A207626 v 1....1....0....2x...x+1..1....N
A207627 u 1....1....0....2x...2x...1....BN
A207628 v 1....1....0....2x...2x...1....BCE
A207629 u 1....1....0....x+1..1....1....CET
A207630 v 1....1....0....x+1..1....1....CO
A207631 u 1....1....0....x+1..x....1....DF
A207632 v 1....1....0....x+1..x....1....DEF
A207633 u 1....1....0....x+1..2x...1....F
A207634 v 1....1....0....x+1..2x...1....F
A207635 u 1....1....0....x+1..x+1..1....DN
A207636 v 1....1....0....x+1..x+1..1....CD
A160232 u 1....x....0....1....2x...0....BCFN
A208341 v 1....x....0....1....2x...0....BCFFN
A085478 u 1....x....0....1....x+1..0....CCOFT*
A078812 v 1....x....0....1....x+1..0....CEFN*
A208342 u 1....x....0....x....x....0....CCFNO
A208343 v 1....x....0....x....x....0....BBCDFZ
A208344 u 1....x....0....x....2x...0....CCFN
A208345 v 1....x....0....x....2x...0....CFZ
A094436 u 1....x....0....x....x+1..0....CFFN
A094437 v 1....x....0....x....x+1..0....CEFF
A117919 u 1....x....0....2x...1....0....BCNT
A135837 v 1....x....0....2x...1....0....BCET
A208328 u 1....x....0....2x...x....0....CCOP
A208329 v 1....x....0....2x...x....0....DPZ
A208330 u 1....x....0....2x...x+1..0....CNPT
A208331 v 1....x....0....2x...x+1..0....CN
A208332 u 1....x....0....2x...2x...0....CCE
A208333 v 1....x....0....2x...2x...0....DZ
A208334 u 1....x....0....x+1..1....0....CCNT
A208335 v 1....x....0....x+1..1....0....CCN*
A208336 u 1....x....0....x+1..x....0....CFNT*
A208337 v 1....x....0....x+1..x....0....ACFN*
A208338 u 1....x....0....x+1..2x...0....CNP
A208339 v 1....x....0....x+1..2x...0....BCNP
A202390 u 1....x....0....x+1..x+1..0....CFPTZ*
A208340 v 1....x....0....x+1..x+1..0....FNPZ*
A208508 u 1....x....0....1....1....1....CCES
A208509 v 1....x....0....1....1....1....BCO
A208510 u 1....x....0....1....x....1....CCCNOS*
A029653 v 1....x....0....1....x....1....BCDOSZ*
A208511 u 1....x....0....1....2x...1....BCFO
A208512 v 1....x....0....1....2x...1....BDFO
A208513 u 1....x....0....1....x+1..1....CCES*
A111125 v 1....x....0....1....x+1..1....COO*
A133567 u 1....x....0....x....1....1....CCOTT
A133084 v 1....x....0....x....1....1....BBCEN
A208514 u 1....x....0....x....x....1....CEFN
A208515 v 1....x....0....x....x....1....BCDFN
A208516 u 1....x....0....x....2x...1....CNN
A208517 v 1....x....0....x....2x...1....CCN
A208518 u 1....x....0....x....x+1..1....CFNT
A208519 v 1....x....0....x....x+1..1....NFFT
A208520 u 1....x....0....2x...1....1....BCTT
A208521 v 1....x....0....2x...1....1....BEN
A208522 u 1....x....0....2x...x....1....CCN
A208523 v 1....x....0....2x...x....1....CCO
A208524 u 1....x....0....2x...x+1..1....CT*
A208525 v 1....x....0....2x...x+1..1....ACNP*
A208526 u 1....x....0....2x...2x...1....CEN
A208527 v 1....x....0....2x...2x...1....CCE
A208606 u 1....x....0....x+1..1....1....CCS
A208607 v 1....x....0....x+1..1....1....CNO
A208608 u 1....x....0....x+1..x....1....CFOT
A208609 v 1....x....0....x+1..x....1....DEN*
A208610 u 1....x....0....x+1..2x...1....CO
A208611 v 1....x....0....x+1..2x...1....DE
A208612 u 1....x....0....x+1..x+1..1....CFNS
A208613 v 1....x....0....x+1..x+1..1....CFN*
A105070 u 1....2x...0....1....1....0....BN
A207536 u 1....2x...0....1....1....0....BCT
A208751 u 1....2x...0....1....x+1..0....CDPT
A208752 v 1....2x...0....1....x+1..0....CNP
A135837 u 1....2x...0....x....1....0....BCNT
A117919 v 1....2x...0....x....1....0....BCNT
A208755 u 1....2x...0....x....x....0....BCDEP
A208756 v 1....2x...0....x....x....0....BCCOZ
A208757 u 1....2x...0....x....2x...0....CDEP
A208758 v 1....2x...0....x....2x...0....CCEPZ
A208763 u 1....2x...0....2x...x....0....CDOP
A208764 v 1....2x...0....2x...x....0....CCCP
A208765 u 1....2x...0....2x...x+1..0....CE
A208766 v 1....2x...0....2x...x+1..0....CC
A208747 u 1....2x...0....2x...2x...0....CDE
A208748 v 1....2x...0....2x...2x...0....CCZ
A208749 u 1....2x...0....x+1..1....0....BCOPT
A208750 v 1....2x...0....x+1..1....0....BCNP*
A208759 u 1....2x...0....x+1..2x....0...CE
A208760 v 1....2x...0....x+1..2x....0...BCO
A208761 u 1....2x...0....x+1..x+1...0...BCCT*
A208762 v 1....2x...0....x+1..x+1...0...BNZ*
A208753 u 1....2x...0....1....1.....1...BCS
A208754 v 1....2x...0....1....1.....1...BO
A105045 u 1....2x...0....1....2x....1...BCCOS*
A208659 v 1....2x...0....1....2x....1...BDOSZ*
A208660 u 1....2x...0....1....x+1...1...CDS
A208904 v 1....2x...0....1....x+1...1...CNO
A208905 u 1....2x...0....x....1.....1...BCT
A208906 v 1....2x...0....x....1.....1...BNN
A208907 u 1....2x...0....x....x.....1...BCN
A208756 v 1....2x...0....x....x.....1...BCCE
A208755 u 1....2x...0....x....2x....1...CEN
A208910 v 1....2x...0....x....2x....1...CCE
A208911 u 1....2x...0....x....x+1...1...BCT
A208912 v 1....2x...0....x....x+1...1...BNT
A208913 u 1....2x...0....2x...1.....1...BCT
A208914 v 1....2x...0....2x...1.....1...BEN
A208915 u 1....2x...0....2x...x.....1...CE
A208916 v 1....2x...0....2x...x.....1...CCO
A208919 u 1....2x...0....2x...x+1...1...CT
A208920 v 1....2x...0....2x...x+1...1...N
A208917 u 1....2x...0....2x...2x....1...CEN
A208918 v 1....2x...0....2x...2x....1...CCNP
A208921 u 1....2x...0....x+1..1.....1...BC
A208922 v 1....2x...0....x+1..1.....1...BON
A208923 u 1....2x...0....x+1..x.....1...BCNO
A208908 v 1....2x...0....x+1..x.....1...BDN*
A208909 u 1....2x...0....x+1..2x....1...BN
A208930 v 1....2x...0....x+1..2x....1...DN
A208931 u 1....2x...0....x+1..x+1...1...BCOS
A208932 v 1....2x...0....x+1..x+1...1...BCO*
A207537 u 1....x+1..0....1....1.....0...BCO
A207538 v 1....x+1..0....1....1.....0...BCE
A122075 u 1....x+1..0....1....x.....0...CCFN*
A037027 v 1....x+1..0....1....x.....0...CCFN*
A209125 u 1....x+1..0....1....2x....0...BCFN*
A164975 v 1....x+1..0....1....2x....0...BF
A209126 u 1....x+1..0....x....x.....0...CDFO*
A209127 v 1....x+1..0....x....x.....0...DFOZ*
A209128 u 1....x+1..0....x....2x....0...CDE*
A209129 v 1....x+1..0....x....2x....0...DEZ
A102756 u 1....x+1..0....x....x+1...0...CFNP*
A209130 v 1....x+1..0....x....x+1...0...CCFNP*
A209131 u 1....x+1..0....2x...x.....0...CDEP*
A209132 v 1....x+1..0....2x...x.....0...CNPZ*
A209133 u 1....x+1..0....2x...2x....0...CDN
A209134 v 1....x+1..0....2x...2x....0...CCN*
A209135 u 1....x+1..0....2x...x+1...0...CN*
A209136 v 1....x+1..0....2x...x+1...0...CCS*
A209137 u 1....x+1..0....x+1..x.....0...CFFP*
A209138 v 1....x+1..0....x+1..x.....0...AFFP*
A209139 u 1....x+1..0....x+1..2x....0...CF*
A209140 v 1....x+1..0....x+1..2x....0...BF
A209141 u 1....x+1..0....x+1..x+1...0...BCF*
A209142 v 1....x+1..0....x+1..x+1...0...BFZ*
A209143 u 1....x+1..0....1....1.....1...CCE*
A209144 v 1....x+1..0....1....1.....1...COO*
A209145 u 1....x+1..0....1....x.....1...CCFN*
A122075 v 1....x+1..0....1....x.....1...CCFN*
A209146 u 1....x+1..0....1....2x....1...BCF*
A209147 v 1....x+1..0....1....2x....1...BF
A209148 u 1....x+1..0....1....x+1...1...CCO*
A209149 v 1....x+1..0....1....x+1...1...CDO*
A209150 u 1....x+1..0....x....1.....1...CCNT*
A208335 v 1....x+1..0....x....1.....1...CDNN*
A209151 u 1....x+1..0....x....x.....1...CFN*
A208337 v 1....x+1..0....x....x.....1...ACFN*
A209152 u 1....x+1..0....x....2x....1...CN*
A208339 v 1....x+1..0....x....x.....1...BCN
A209153 u 1....x+1..0....x....x+1...1...CFT*
A208340 v 1....x+1..0....x....x.....1...FNZ*
A209154 u 1....x+1..0....2x...1.....1...BCT*
A209157 v 1....x+1..0....2x...1.....1...BNN
A209158 u 1....x+1..0....2x...x.....1...CN*
A209159 v 1....x+1..0....2x...x.....1...CO*
A209160 u 1....x+1..0....2x...2x....1...CN*
A209161 v 1....x+1..0....2x...2x....1...CE
A209162 u 1....x+1..0....2x...x+1...1...CT*
A209163 v 1....x+1..0....2x...x+1...1...CO*
A209164 u 1....x+1..0....x+1..1.....1...CC*
A209165 v 1....x+1..0....x+1..1.....1...CCN
A209166 u 1....x+1..0....x+1..x.....1...CFF*
A209167 v 1....x+1..0....x+1..x.....1...FF*
A209168 u 1....x+1..0....x+1..2x....1...CF*
A209169 v 1....x+1..0....x+1..2x....1...CF
A209170 u 1....x+1..0....x+1..x+1...1...CF*
A209171 v 1....x+1..0....x+1..x+1...1...CF*
A053538 u x....1....0....1....1.....0...BBCCFN
A076791 v x....1....0....1....1.....0...BBCDF
A209172 u x....1....0....1....2x....0...BCCFF
A209413 v x....1....0....1....2x....0...BCCFF
A094441 u x....1....0....1....x+1...0...CFFFN
A094442 v x....1....0....1....x+1...0...CEFFF
A054142 u x....1....0....x....x+1...0...CCFOT*
A172431 v x....1....0....x....x+1...0...CEFN*
A008288 u x....1....0....2x...1.....0...CCOO*
A035607 v x....1....0....2x...1.....0...ACDE*
A209414 u x....1....0....2x...x+1...0...CCS
A112351 v x....1....0....2x...x+1...0...CON
A209415 u x....1....0....x+1..x.....0...CCTN
A209416 v x....1....0....x+1..x.....0...ACN*
A209417 u x....1....0....x+1..2x....0...CC
A209418 v x....1....0....x+1..2x....0...BBC
A209419 u x....1....0....x+1..x+1...0...CFTZ*
A209420 v x....1....0....x+1..x+1...0...FNZ*
A209421 u x....1....0....1....1.....1...CCN
A209422 v x....1....0....1....1.....1...CD
A209555 u x....1....0....1....x.....1...CNN
A209556 v x....1....0....1....x.....1...CNN
A209557 u x....1....0....1....2x....1...BCN
A209558 v x....1....0....1....2x....1...BN
A209559 u x....1....0....1....x+1...1...CN
A209560 v x....1....0....1....x+1...1...CN
A209561 u x....1....0....x....1.....1...CCNNT*
A209562 v x....1....0....x....1.....1...CDNNT*
A209563 u x....1....0....x....x.....1...CCFT^
A209564 v x....1....0....x....x.....1...CFN^
A209565 u x....1....0....x....2x....1...CC^
A209566 v x....1....0....x....2x....1...BC^
A209567 u x....1....0....x....x+1...1...CNT*
A209568 v x....1....0....x....x+1...1...NNS*
A209569 u x....1....0....2x...1.....1...CNO*
A209570 v x....1....0....2x...1.....1...DNN*
A209571 u x....1....0....2x...x.....1...CCS^
A209572 v x....1....0....2x...x.....1...CN^
A209573 u x....1....0....2x...x+1...1...CNS
A209574 v x....1....0....2x...x+1...1...NO
A209575 u x....1....0....2x...2x....1...CC
A209576 v x....1....0....2x...2x....1...C
A209577 u x....1....0....x+1..1.....1...CNNT
A209578 v x....1....0....x+1..1.....1...CNN
A209579 u x....1....0....x+1..x.....1...CNNT
A209580 v x....1....0....x+1..x.....1...NN*
A209581 u x....1....0....x+1..2x....1...CN
A209582 v x....1....0....x+1..2x....1...BN
A209583 u x....1....0....x+1..x+1...1...CT*
A209584 v x....1....0....x+1..x+1...1...CN*
A121462 u x....x....0....x....x+1...0...BCFFNZ
A208341 v x....x....0....x....x+1...0...BCFFN
A209687 u x....x....0....2x...x+1...0...BCNZ
A208339 v x....x....0....2x...x+1...0...BCN
A115241 u x....x....0....1....1.....1...CDNZ*
A209688 v x....x....0....1....1.....1...DDN*
A209689 u x....x....0....1....x.....1...FNZ^
A209690 v x....x....0....1....x.....1...FN^
A209691 u x....x....0....1....2x....1...BCZ^
A209692 v x....x....0....1....2x....1...BCC^
A209693 u x....x....0....1....x+1...1...NNZ*
A209694 v x....x....0....1....x+1...1...CN*
A209697 u x....x....0....x....x+1...1...BNZ
A209698 v x....x....0....x....x+1...1...BNT
A209699 u x....x....0....2x...1.....1...BNNZ
A209700 v x....x....0....2x...1.....1...BDN
A209701 u x....x....0....2x...x+1...1...NZ
A209702 v x....x....0....2x...x+1...1...N
A209703 u x....x....0....x+1..1.....1...FNTZ
A209704 v x....x....0....x+1..1.....1...FNNT
A209705 u x....x....0....x+1..x+1...1...BNZ*
A209706 v x....x....0....x+1..x+1...1...BCN*
A209695 u x....x+1..0....2x...x+1...0...ACN*
A209696 v x....x+1..0....2x...x+1...0...CDN*
A209830 u x....x+1..0....x+1..2x....0...ACF
A209831 v x....x+1..0....x+1..2x....0...BCF*
A209745 u x....x+1..0....x+1..x+1...0...ABF*
A209746 v x....x+1..0....x+1..x+1...0...BFZ*
A209747 u x....x+1..0....1....1.....1...ADE*
A209748 v x....x+1..0....1....1.....1...DEO
A209749 u x....x+1..0....1....x.....1...ANN*
A209750 v x....x+1..0....1....x.....1...CNO
A209751 u x....x+1..0....1....2x....1...ABN*
A209752 v x....x+1..0....1....2x....1...BN
A209753 u x....x+1..0....1....x+1...1...AN*
A209754 v x....x+1..0....1....x+1...1...NT*
A209755 u x....x+1..0....x....1.....1...AFN
A209756 v x....x+1..0....x....1.....1...FNO*
A209759 u x....x+1..0....x....2x....1...ACF^
A209760 v x....x+1..0....x....2x....1...CF^*
A209761 u x....x+1..0....x.....x+1..1...ABNS*
A209762 v x....x+1..0....x.....x+1..1...BNS*
A209763 u x....x+1..0....2x....1....1...ABN*
A209764 v x....x+1..0....2x....1....1...BNN
A209765 u x....x+1..0....2x....x....1...ACF^*
A209766 v x....x+1..0....2x....x....1...CF^
A209767 u x....x+1..0....2x....x+1..1...AN*
A209768 v x....x+1..0....2x....x+1..1...N*
A209769 u x....x+1..0....x+1...1....1...AF*
A209770 v x....x+1..0....x+1...1....1...FN
A209771 u x....x+1..0....x+1...x....1...ABN*
A209772 v x....x+1..0....x+1...x....1...BN*
A209773 u x....x+1..0....x+1...2x...1...AF
A209774 v x....x+1..0....x+1...2x...1...FN*
A209775 u x....x+1..0....x+1...x+1..1...AB*
A209776 v x....x+1..0....x+1...x+1..1...BC*
A210033 u 1....1....1....1.....x....1...BCN
A210034 v 1....1....1....1.....x....1...BCDFN
A210035 u 1....1....1....1.....2x...1...BBF
A210036 v 1....1....1....1.....2x...1...BBFF
A210037 u 1....1....1....1.....x+1..1...BCFFN
A210038 v 1....1....1....1.....x+1..1...BCFFN
A210039 u 1....1....1....x.....1....1...BCOT
A210040 v 1....1....1....x.....1....1...BCEN
A210042 u 1....1....1....x.....x....1...BCDEOT*
A124927 v 1....1....1....x.....x....1...BCDET*
A210041 u 1....1....1....x.....2x...1...BFO
A209758 v 1....1....1....x.....2x...1...BCFO
A210187 u 1....1....1....x.....x+1..1...DTF*
A210188 v 1....1....1....x.....x+1..1...DNF*
A210189 u 1....1....1....2x....1....1...BT
A210190 v 1....1....1....2x....1....1...BN
A210191 u 1....1....1....2x....x....1...CO*
A210192 v 1....1....1....2x....x....1...CCO*
A210193 u 1....1....1....2x....x+1..1...CPT
A210194 v 1....1....1....2x....x+1..1...CN
A210195 u 1....1....1....2x....2x...1...BOPT*
A210196 v 1....1....1....2x....2x...1...BCC*
A210197 u 1....1....1....x+1...1....1...BCOT
A210198 v 1....1....1....x+1...1....1...BCEN
A210199 u 1....1....1....x+1...x....1...DFT
A210200 v 1....1....1....x+1...x....1...DFO*
A210201 u 1....1....1....x+1...2x...1...BFP
A210202 v 1....1....1....x+1...2x...1...BF
A210203 u 1....1....1....x+1...x+1..1...BDOP
A210204 v 1....1....1....x+1...x+1..1...BCDN*
A210211 u x....1....1....1.....2x...1...BCFN
A210212 v x....1....1....1.....2x...1...BFN
A210213 u x....1....1....1.....x+1..1...CFFN
A210214 v x....1....1....1.....x+1..1...CFFO
A210215 u x....1....1....x.....x....1...BCDFT^
A210216 v x....1....1....x.....x....1...BCFO^
A210217 u x....1....1....x.....2x...1...CDF^
A210218 v x....1....1....x.....2x...1...BCF^
A210219 u x....1....1....x.....x+1..1...CNSTF*
A210220 v x....1....1....x.....x+1..1...FNNT*
A104698 u x....1....1....2x......1..1...CENS*
A210220 v x....1....1....2x....x+1..1...DNNT*
A210223 u x....1....1....2x....x....1...CD^
A210224 v x....1....1....2x....x....1...CO^
A210225 u x....1....1....2x....x+1..1...CNP
A210226 v x....1....1....2x....x+1..1...NOT
A210227 u x....1....1....2x....2x...1...CDP^
A210228 v x....1....1....2x....2x...1...C^
A210229 u x....1....1....x+1...1....1...CFNN
A210230 v x....1....1....x+1...1....1...CCN
A210231 u x....1....1....x+1...x....1...CNT
A210232 v x....1....1....x+1...x....1...NN*
A210233 u x....1....1....x+1...2x...1...CNP
A210234 v x....1....1....x+1...2x...1...BN
A210235 u x....1....1....x+1...x+1..1...CCFPT*
A210236 v x....1....1....x+1...x+1..1...CFN*
A124927 u x....x....1....1.....1....1...BCDEET*
A210042 v x....1....1....x+1...x+1..1...BDEOT*
A210216 u x....x....1....1.....x....1...BCFO^
A210215 v x....x....1....1.....x....1...BCDFT^
A210549 u x....x....1....1.....2x...1...BCF^
A210550 v x....x....1....1.....2x...1...BDF^
A172431 u x....x....1....1.....x+1..1...CEFN*
A210551 v x....x....1....1.....x+1..1...CFOT*
A210552 u x....x....1....x.....1....1...BBCFNO
A210553 v x....x....1....x.....1....1...BNNFB
A208341 u x....x....1....x.....x+1..1...BCFFN
A210554 v x....x....1....x.....x+1..1...BNFFT
A210555 u x....x....1....2x....1....1...BCNN
A210556 v x....x....1....2x....1....1...BENP
A210557 u x....x....1....2x....x+1..1...CNP
A210558 v x....x....1....2x....x+1..1...N
A210559 u x....x....1....x+1...1....1...CEF
A210560 v x....x....1....x+1...1....1...OFNS
A210561 u x....x....1....x+1...x....1...BCNP^
A210562 v x....x....1....x+1...x....1...BDP*^
A210563 u x....x....1....x+1...2x...1...CFP^
A210564 v x....x....1....x+1...2x...1...DF^
A013609 u x....x....1....x+1...x+1..1...BCEPT*
A209757 v x....x....1....x+1...x+1..1...BCOS*
A209819 u x....2x...1....x+1...x....1...CFN^
A209820 v x....2x...1....x+1...x....1...DF^
A209996 u x....2x...1....x+1...2x...1...CP^
A209998 v x....2x...1....x+1...2x...1...DP^
A209999 u x....x+1..1....1.....x+1..1...FN*
A210287 v x....x+1..1....1.....x+1..1...CFT*
A210565 u x....x+1..1....x.....1....1...FNT*
A210595 v x....x+1..1....x.....1....1...FNNT
A210598 u x....x+1..1....x+1...2x...1...FN*
A210599 v x....x+1..1....x+1...2x...1...FN
A210600 u x....x+1..1....x+1...x+1..1...BF*
A210601 v x....x+1..1....x+1...x+1..1...BF*
A210597 u 2x...1....1....x+1...1....1...BF
A210601 v 2x...1....1....x+1...1....1...BFN*
A210603 u 2x...1....1....x+1...x+1..1...BF
A210738 v 2x...1....1....x+1...x+1..1...CBF*
A210739 u 2x...x....1....x+1...x....1...CF^
A210740 v 2x...x....1....x+1...x....1...DF*^
A210741 u 2x...x....1....x+1...x+1..1...BCFO
A210742 v 2x...x....1....x+1...x+1..1...CFO*
A210743 u 2x...x+1..1....x+1...1....1...F
A210744 v 2x...x+1..1....x+1...1....1...FN
A210747 u 2x...x+1..1....x+1...x+1..1...FF
A210748 v 2x...x+1..1....x+1...x+1..1...CFF*
A210749 u x+1..1....1....x+1...2x...1...BCF
A210750 v x+1..1....1....x+1...2x...1...BF
A210751 u x+1..x....1....x+1...2x...1...FNT
A210752 v x+1..x....1....x+1...2x...1...FN
A210753 u x+1..x....1....x+1...x+1..1...BNZ*
A210754 v x+1..x....1....x+1...x+1..1...BCT*
A210755 u x+1..2x...1....x+1...x+1..1...N*
A210756 v x+1..2x...1....x+1...x+1..1...CT*
A210789 u 1....x....0....x+2...x-1..0...CFFN
A210790 v 1....x....0....x+2...x-1..0...CEFF
A210791 u 1....x....0....x-1...x+2..0...CFNP
A210792 v 1....x....0....x-1...x+2..0...CF
A210793 u 1....x+1..0....x+2...x-1..0...CFNP
A210794 v 1....x+1..0....x+2...x-1..0...FPP
A210795 u 1....x....1....x+2...x-1..0...FN
A210796 v 1....x....1....x+2...x-1..0...FO
A210797 u 1....x....0....x+2...x-1..1...CF
A210798 v 1....x....0....x+2...x-1..1...F
A210799 u 1....x+1..1....x+2...x-1..0...FN
A210800 v 1....x+1..1....x+2...x-1..0...F
A210801 u 1....x+1..1....x+2...x-1..1...FN
A210802 v 1....x+1..1....x+2...x-1..1...F
A210803 u 1....x....0....x-1...x+3..0...F*
A210804 v 1....x....0....x-1...x+3..0...F*
A210805 u 1....x....0....x+2...x-1.-1...CFFN
A210806 v 1....x....0....x+2...x-1.-1...FF
A210858 u 1....x....0....x+n...x....0...CFT*
A210859 v 1....x....0....x+n...x....0...FN*
A210860 u 1....x+1..0....x+n...x....0...F
A210861 v 1....x+1..0....x+n...x....0...F*
A210862 u 1....x....1....x+n-1.x....0...FN
A210863 v 1....x....1....x+n-1.x....0...FS
A210864 u 1....x....1....x+n...x....0...FN
A210865 v 1....x....1....x+n...x....0...FT
A210866 u 1....x....0....x+n...x...-x...CFT
A210867 v 1....x....0....x+n...x...-x...FN
A210868 u 1....x....0....x+1...x-1..0...BCFN
A210869 v 1....x....0....x+1...x-1..0...BBCFNZ
A210870 u 1....x....0....x+1...x-1..1...CFFN
A210871 v 1....x....0....x+1...x-1..1...CFF
A210872 u x....1...-1....x.....x....1...BDFZ^
A210873 v x....1...-1....x.....x....1...BCFN^
A210876 u x....1....1....x.....x....x...BCCF^
A210877 v x....1....1....x.....x....x...BDFNZ^
A210878 u x....2x...0....x+1...x....1...DFZ^
A210879 v x....2x...0....x+1...x....1...FC*^
Some of these triangles have irregular row lengths, making it difficult to retrieve individual rows/columns/diagonals without actually computing the recurrence. - Georg Fischer, Sep 04 2021

Examples

			First five rows:
1
1...1
1...3...1
1...5...4...1
1...7...9...5...1
First five polynomials u(n,x):
1
1 + x
1 + 3x + x^2
1 + 5x + 4x^2 + x^3
1 + 7x + 9x^2 + 5x^3 + x^4

Crossrefs

Cf. A029653, A208508, A208509.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A029653 *)
  • Python
    from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017

Formula

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also, u(n,x)=(x+1)*u(n-1,x)+x for n>2, with u(n,2)=x+1.

Extensions

Corrected by Philippe Deléham, Apr 10 2012
Corrections and additions by Clark Kimberling, May 09 2012
Corrections in the overview by Georg Fischer, Sep 04 2021

A000204 Lucas numbers (beginning with 1): L(n) = L(n-1) + L(n-2) with L(1) = 1, L(2) = 3.

Original entry on oeis.org

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498, 3010349, 4870847, 7881196, 12752043, 20633239, 33385282, 54018521, 87403803, 141422324
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

See A000032 for the version beginning 2, 1, 3, 4, 7, ...
Also called Schoute's accessory series (see Jean, 1984). - N. J. A. Sloane, Jun 08 2011
L(n) is the number of matchings in a cycle on n vertices: L(4)=7 because the matchings in a square with edges a, b, c, d (labeled consecutively) are the empty set, a, b, c, d, ac and bd. - Emeric Deutsch, Jun 18 2001
This comment covers a family of sequences which satisfy a recurrence of the form a(n) = a(n-1) + a(n-m), with a(n) = 1 for n = 1..m - 1, a(m) = m + 1. The generating function is (x + m*x^m)/(1 - x - x^m). Also a(n) = 1 + n*Sum_{i=1..n/m} binomial(n - 1 - (m - 1)*i, i - 1)/i. This gives the number of ways to cover (without overlapping) a ring lattice (or necklace) of n sites with molecules that are m sites wide. Special cases: m = 2: A000204, m = 3: A001609, m = 4: A014097, m = 5: A058368, m = 6: A058367, m = 7: A058366, m = 8: A058365, m = 9: A058364.
L(n) is the number of points of period n in the golden mean shift. The number of orbits of length n in the golden mean shift is given by the n-th term of the sequence A006206. - Thomas Ward, Mar 13 2001
Row sums of A029635 are 1, 1, 3, 4, 7, ... - Paul Barry, Jan 30 2005
a(n) counts circular n-bit strings with no repeated 1's. E.g., for a(5): 00000 00001 00010 00100 00101 01000 01001 01010 10000 10010 10100. Note #{0...} = fib(n+1), #{1...} = fib(n-1), #{000..., 001..., 100...} = a(n-1), #{010..., 101...} = a(n-2). - Len Smiley, Oct 14 2001
Row sums of the triangle in A182579. - Reinhard Zumkeller, May 07 2012
If p is prime then L(p) == 1 (mod p). L(2^k) == -1 (mod 2^(k+1)) for k = 0,1,2,... - Thomas Ordowski, Sep 25 2013
Satisfies Benford's law [Brown-Duncan, 1970; Berger-Hill, 2017]. - N. J. A. Sloane, Feb 08 2017

Examples

			G.f. = x + 3*x^2 + 4*x^3 + 7*x^4 + 11*x^5 + 18*x^6 + 29*x^7 + 47*x^8 + ...

References

  • P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 69.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 46.
  • Leonhard Euler, Introductio in analysin infinitorum (1748), sections 216 and 229.
  • G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 148.
  • Silvia Heubach and Toufik Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
  • V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
  • R. V. Jean, Mathematical Approach to Pattern and Form in Plant Growth, Wiley, 1984. See p. 5. - N. J. A. Sloane, Jun 08 2011
  • Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.

Links

Crossrefs

Cf. A000032, A000045, A061084, A027960, A001609, A014097, A000079, A003269, A003520, A005708, A005709, A005710, A006206, A101033, A101032, A100492, A099731, A094216, A094638, A000108, A090946 (complement).

Programs

  • Haskell
    a000204 n = a000204_list !! n
a000204_list = 1 : 3 : zipWith (+) a000204_list (tail a000204_list)
-- Reinhard Zumkeller, Dec 18 2011
  • Magma
    [Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 17 2017
  • Maple
    A000204 := proc(n) option remember; if n <=2 then 2*n-1; else procname(n-1)+procname(n-2); fi; end;
with(combinat): A000204 := n->fibonacci(n+1)+fibonacci(n-1);
# alternative Maple program:
L:= n-> (<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1]:
seq(L(n), n=1..50);  # Alois P. Heinz, Jul 25 2008
# Alternative:
a := n -> `if`(n=1, 1, `if`(n=2, 3, hypergeom([(1-n)/2, -n/2], [1-n], -4))):
seq(simplify(a(n)), n=1..39); # Peter Luschny, Sep 03 2019
  • Mathematica
    c = (1 + Sqrt[5])/2; Table[Expand[c^n + (1 - c)^n], {n, 30}] (* Artur Jasinski, Oct 05 2008 *)
Table[LucasL[n, 1], {n, 36}] (* Zerinvary Lajos, Jul 09 2009 *)
LinearRecurrence[{1, 1},{1, 3}, 50] (* Sture Sjöstedt, Nov 28 2011 *)
lukeNum[n_] := If[n < 1, 0, LucasL[n]]; (* Michael Somos, May 18 2015 *)
lukeNum[n_] := SeriesCoefficient[x D[Log[1 / (1 - x - x^2)], x], {x, 0, n}]; (* Michael Somos, May 18 2015 *)
  • PARI
    A000204(n)=fibonacci(n+1)+fibonacci(n-1) \\ Michael B. Porter, Nov 05 2009
  • Python
    from functools import cache
@cache
def a(n): return [1, 3][n-1] if n < 3 else a(n-1) + a(n-2)
print([a(n) for n in range(1, 41)]) # Michael S. Branicky, Nov 13 2022
  • Python
    [(i:=-1)+(j:=2)] + [(j:=i+j)+(i:=j-i) for  in range(100)] # _Jwalin Bhatt, Apr 02 2025
  • Sage
    def A000204():
    x, y = 1, 2
    while true:
       yield x
       x, y = x + y, x
a = A000204(); print([next(a) for i in range(39)])  # Peter Luschny, Dec 17 2015
  • Scala
    def lucas(n: BigInt): BigInt = {
  val zero = BigInt(0)
  def fibTail(n: BigInt, a: BigInt, b: BigInt): BigInt = n match {
    case `zero` => a
    case _ => fibTail(n - 1, b, a + b)
  }
  fibTail(n, 2, 1)
}
(1 to 50).map(lucas()) // _Alonso del Arte, Oct 20 2019

Formula

Expansion of x(1 + 2x)/(1 - x - x^2). - Simon Plouffe, dissertation 1992; multiplied by x. - R. J. Mathar, Nov 14 2007
a(n) = A000045(2n)/A000045(n). - Benoit Cloitre, Jan 05 2003
For n > 1, L(n) = F(n + 2) - F(n - 2), where F(n) is the n-th Fibonacci number (A000045). - Gerald McGarvey, Jul 10 2004
a(n+1) = 4*A054886(n+3) - A022388(n) - 2*A022120(n+1) (a conjecture; note that the above sequences have different offsets). - Creighton Dement, Nov 27 2004
a(n) = Sum_{k=0..floor((n+1)/2)} (n+1)*binomial(n - k + 1, k)/(n - k + 1). - Paul Barry, Jan 30 2005
L(n) = A000045(n+3) - 2*A000045(n). - Creighton Dement, Oct 07 2005
L(n) = A000045(n+1) + A000045(n-1). - John Blythe Dobson, Sep 29 2007
a(n) = 2*Fibonacci(n-1) + Fibonacci(n), n >= 1. - Zerinvary Lajos, Oct 05 2007
L(n) is the term (1, 1) in the 1 X 2 matrix [2, -1].[1, 1; 1, 0]^n. - Alois P. Heinz, Jul 25 2008
a(n) = phi^n + (1 - phi)^n = phi^n + (-phi)^(-n) = ((1 + sqrt(5))^n + (1 - sqrt(5))^n)/(2^n) where phi is the golden ratio (A001622). - Artur Jasinski, Oct 05 2008
a(n) = A014217(n+1) - A014217(n-1). See A153263. - Paul Curtz, Dec 22 2008
a(n) = ((1 + sqrt(5))^n - (1 - sqrt(5))^n)/(2^n*sqrt(5)) + ((1 + sqrt(5))^(n - 1) - (1 - sqrt(5))^(n - 1))/(2^(n - 2)*sqrt(5)). - Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009, Jan 14 2009
From Hieronymus Fischer, Oct 20 2010 (Start)
Continued fraction for n odd: [L(n); L(n), L(n), ...] = L(n) + fract(Fib(n) * phi).
Continued fraction for n even: [L(n); -L(n), L(n), -L(n), L(n), ...] = L(n) - 1 + fract(Fib(n)*phi). Also: [L(n) - 2; 1, L(n) - 2, 1, L(n) - 2, ...] = L(n) - 2 + fract(Fib(n)*phi). (End)
INVERT transform of (1, 2, -1, -2, 1, 2, ...). - Gary W. Adamson, Mar 07 2012
L(2n - 1) = floor(phi^(2n - 1)); L(2n) = ceiling(phi^(2n)). - Thomas Ordowski, Jun 15 2012
a(n) = hypergeom([(1 - n)/2, -n/2], [1 - n], -4) for n >= 3. - Peter Luschny, Sep 03 2019
E.g.f.: 2*(exp(x/2)*cosh(sqrt(5)*x/2) - 1). - Stefano Spezia, Jul 26 2022

Extensions

Additional comments from Yong Kong (ykong(AT)curagen.com), Dec 16 2000
Plouffe Maple line edited by N. J. A. Sloane, May 13 2008

A003945 Expansion of g.f. (1+x)/(1-2*x).

Original entry on oeis.org

1, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472, 6442450944, 12884901888
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Coordination sequence for infinite tree with valency 3.
Number of Hamiltonian cycles in K_3 X P_n.
Number of ternary words of length n avoiding aa, bb, cc.
For n > 0, row sums of A029635. - Paul Barry, Jan 30 2005
Binomial transform is {1, 4, 13, 40, 121, 364, ...}, see A003462. - Philippe Deléham, Jul 23 2005
Convolved with the Jacobsthal sequence A001045 = A001786: (1, 4, 12, 32, 80, ...). - Gary W. Adamson, May 23 2009
Equals (n+1)-th row sums of triangle A161175. - Gary W. Adamson, Jun 05 2009
a(n) written in base 2: a(0) = 1, a(n) for n >= 1: 11, 110, 11000, 110000, ..., i.e.: 2 times 1, (n-1) times 0 (see A003953(n)). - Jaroslav Krizek, Aug 17 2009
INVERTi transform of A003688. - Gary W. Adamson, Aug 05 2010
An elephant sequence, see A175655. For the central square four A[5] vectors, with decimal values 42, 138, 162 and 168, lead to this sequence. For the corner squares these vectors lead to the companion sequence A083329. - Johannes W. Meijer, Aug 15 2010
A216022(a(n)) != 2 and A216059(a(n)) != 3. - Reinhard Zumkeller, Sep 01 2012
Number of length-n strings of 3 letters with no two adjacent letters identical. The general case (strings of r letters) is the sequence with g.f. (1+x)/(1-(r-1)*x). - Joerg Arndt, Oct 11 2012
Sums of pairs of rows of Pascal's triangle A007318, T(2n,k)+T(2n+1,k); Sum_{n>=1} A000290(n)/a(n) = 4. - John Molokach, Sep 26 2013

Links

Crossrefs

Essentially same as A007283 (3*2^n) and A042950.
Cf. A001787, A001045, A161175.
Generating functions of the form (1+x)/(1-k*x) for k=1 to 12: A040000, A003945, A003946, A003947, A003948, A003949, A003950, A003951, A003952.
Generating functions of the form (1+x)/(1-k*x) for k=13 to 30: A170732, A170733, A170734, A170735, A170736, A170737, A170738, A170739, A170740, A170741, A170742, A170743, A170744, A170745, A170746, A170747, A170748.
Generating functions of the form (1+x)/(1-k*x) for k=31 to 50: A170749, A170750, A170751, A170752, A170753, A170754, A170755, A170756, A170757, A170758, A170759, A170760, A170761, A170762, A170763, A170764, A170765, A170766, A170767, A170768, A170769.
Cf. A003688.

Programs

  • Maple
    k := 3; if n = 0 then 1 else k*(k-1)^(n-1); fi;
  • Mathematica
    Join[{1}, 3*2^Range[0, 60]] (* Vladimir Joseph Stephan Orlovsky, Jun 09 2011 *)
Table[2^n+Floor[2^(n-1)], {n,0,30}] (* Martin Grymel, Oct 17 2012 *)
CoefficientList[Series[(1+x)/(1-2x),{x,0,40}],x] (* or *) LinearRecurrence[ {2},{1,3},40] (* Harvey P. Dale, May 04 2017 *)
  • PARI
    a(n)=if(n,3<Charles R Greathouse IV, Jan 12 2012

Formula

a(0) = 1; for n > 0, a(n) = 3*2^(n-1).
a(n) = 2*a(n-1), n > 1; a(0)=1, a(1)=3.
More generally, the g.f. (1+x)/(1-k*x) produces the sequence [1, 1 + k, (1 + k)*k, (1 + k)*k^2, ..., (1+k)*k^(n-1), ...], with a(0) = 1, a(n) = (1+k)*k^(n-1) for n >= 1. Also a(n+1) = k*a(n) for n >= 1. - Zak Seidov and N. J. A. Sloane, Dec 05 2009
The g.f. (1+x)/(1-k*x) produces the sequence with closed form (in PARI notation) a(n)=(n>=0)*k^n+(n>=1)*k^(n-1). - Jaume Oliver Lafont, Dec 05 2009
Binomial transform of A000034. a(n) = (3*2^n - 0^n)/2. - Paul Barry, Apr 29 2003
a(n) = Sum_{k=0..n} (n+k)*binomial(n, k)/n. - Paul Barry, Jan 30 2005
a(n) = Sum_{k=0..n} A029653(n, k)*x^k for x = 1. - Philippe Deléham, Jul 10 2005
Binomial transform of A000034. Hankel transform is {1,-3,0,0,0,...}. - Paul Barry, Aug 29 2006
a(0) = 1, a(n) = 2 + Sum_{k=0..n-1} a(k) for n >= 1. - Joerg Arndt, Aug 15 2012
a(n) = 2^n + floor(2^(n-1)). - Martin Grymel, Oct 17 2012
E.g.f.: (3*exp(2*x) - 1)/2. - Stefano Spezia, Jan 31 2023

Extensions

Edited by N. J. A. Sloane, Dec 04 2009

A034807 Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.

Original entry on oeis.org

2, 1, 1, 2, 1, 3, 1, 4, 2, 1, 5, 5, 1, 6, 9, 2, 1, 7, 14, 7, 1, 8, 20, 16, 2, 1, 9, 27, 30, 9, 1, 10, 35, 50, 25, 2, 1, 11, 44, 77, 55, 11, 1, 12, 54, 112, 105, 36, 2, 1, 13, 65, 156, 182, 91, 13, 1, 14, 77, 210, 294, 196, 49, 2, 1, 15, 90, 275, 450, 378, 140, 15, 1, 16, 104
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

These polynomials arise in the following setup. Suppose G and H are power series satisfying G + H = G*H = 1/x. Then G^n + H^n = (1/x^n)*L_n(-x).
Apart from signs, triangle of coefficients when 2*cos(nt) is expanded in terms of x = 2*cos(t). For example, 2*cos(2t) = x^2 - 2, 2*cos(3t) = x^3 - 3x and 2*cos(4t) = x^4 - 4x^2 + 2. - Anthony C Robin, Jun 02 2004
Triangle of coefficients of expansion of Z_{nk} in terms of Z_k.
Row n has 1 + floor(n/2) terms. - Emeric Deutsch, Dec 25 2004
T(n,k) = number of k-matchings of the cycle C_n (n > 1). Example: T(6,2)=9 because the 2-matchings of the hexagon with edges a, b, c, d, e, f are ac, ad, ae, bd, be, bf, ce, cf and df. - Emeric Deutsch, Dec 25 2004
An example for the first comment: G=c(x), H=1/(x*c(x)) with c(x) the o.g.f. Catalan numbers A000108: (x*c(x))^n + (1/c(x))^n = L(n,-x)= Sum_{k=0..floor(n/2)} T(n,k)*(-x)^k.
This triangle also supplies the absolute values of the coefficients in the multiplication formulas for the Lucas numbers A000032.
From L. Edson Jeffery, Mar 19 2011: (Start)
This sequence is related to rhombus substitution tilings. A signed version of it (see A132460), formed as a triangle with interlaced zeros extending each row to n terms, begins as
{2}
{1, 0}
{1, 0, -2}
{1, 0, -3, 0}
{1, 0, -4, 0, 2}
{1, 0, -5, 0, 5, 0}
....
For the n X n tridiagonal unit-primitive matrix G_(n,1) (n >= 2) (see the L. E. Jeffery link below), defined by
G_(n,1) =
(0 1 0 ... 0)
(1 0 1 0 ... 0)
(0 1 0 1 0 ... 0)
...
(0 ... 0 1 0 1)
(0 ... 0 2 0),
Row n (i.e., {T(n,k)}, k=0..n) of the signed table gives the coefficients of its characteristic function: c_n(x) = Sum_{k=0..n} T(n,k)*x^(n-k) = 0. For example, let n=3. Then
G_(3,1) =
(0 1 0)
(1 0 1)
(0 2 0),
and row 3 of the table is {1,0,-3,0}. Hence c_3(x) = x^3 - 3*x = 0. G_(n,1) has n distinct eigenvalues (the solutions of c_n(x) = 0), given by w_j = 2*cos((2*j-1)*Pi/(2*n)), j=1..n. (End)
For n > 0, T(n,k) is the number of k-subsets of {1,2,...,n} which contain neither consecutive integers nor both 1 and n. Equivalently, T(n,k) is the number of k-subsets without neighbors of a set of n points on a circle. - José H. Nieto S., Jan 17 2012
With the first column omitted, this gives A157000. - Philippe Deléham, Mar 17 2013
The number of necklaces of k black and n - k white beads with no adjacent black beads (Kaplansky 1943). Coefficients of the Dickson polynomials D(n,x,-a). - Peter Bala, Mar 09 2014
From Tom Copeland, Nov 07 2015: (Start)
This triangular array is composed of interleaved rows of reversed, unsigned A127677 (cf. A156308, A217476, A263916) and reversed A111125 (cf. A127672).
See also A113279 for another connection to symmetric and Faber polynomials.
The difference of consecutive rows gives the previous row shifted.
For relations among the characteristic polynomials of Cartan matrices of the Coxeter root groups, Chebyshev polynomials, cyclotomic polynomials, and the polynomials of this entry, see Damianou (p. 12, 20, and 21) and Damianou and Evripidou (p. 7). (End)
Diagonals are related to multiplicities of eigenvalues of the Laplacian on hyperspheres through A029635. - Tom Copeland, Jan 10 2016
For n>=3, also the independence and matching polynomials of the n-cycle graph C_n. See also A284966. - Eric W. Weisstein, Apr 06 2017
Apparently, with the rows aerated and then the 2s on the diagonal removed, this matrix becomes the reverse, or mirror, of unsigned A117179. See also A114525 - Tom Copeland, May 30 2017
Briggs's (1633) table with an additional column of 2s on the right can be used to generate this table. See p. 69 of the Newton reference. - Tom Copeland, Jun 03 2017
From Liam Solus, Aug 23 2018: (Start)
For n>3 and k>0, T(n,k) equals the number of Markov equivalence classes with skeleton the cycle on n nodes having exactly k immoralities. See Theorem 2.1 of the article by A. Radhakrishnan et al. below.
For n>2 odd and r = floor(n/2)-1, the n-th row is the coefficient vector of the Ehrhart h*-polynomial of the r-stable (n,2)-hypersimplex. See Theorem 4.14 in the article by B. Braun and L. Solus below.
(End)
Conjecture: If a(n) = H(a,b,c,d,n) is a second-order linear recurrence with constant coefficients defined as a(0) = a, a(1)= b, a(n) = c*a(n-1) + d*a(n-2) then a(m*n) = H(a, H(a,b,c,d,m), Sum_{k=0..floor(m/2)} T(m,k)*c^(m-2*k)*d^k, (-1)^(m+1)*d^m, n) (Wolfdieter Lang). - Gary Detlefs, Feb 06 2023
For the proof of the preceding conjecture see the Detlefs and Lang link. There also proofs for several properties of this table are found. - Wolfdieter Lang, Apr 25 2023
From Mohammed Yaseen, Nov 09 2024: (Start)
Let m - 1/m = x, then
m^2 + 1/m^2 = x^2 + 2,
m^3 - 1/m^3 = x^3 + 3*x,
m^4 + 1/m^4 = x^4 + 4*x^2 + 2,
m^5 - 1/m^5 = x^5 + 5*x^3 + 5*x,
m^6 + 1/m^6 = x^6 + 6*x^4 + 9*x^2 + 2,
m^7 - 1/m^7 = x^7 + 7*x^5 + 14*x^3 + 7*x, etc. (End)

Examples

			I have seen two versions of these polynomials: One version begins L_0 = 2, L_1 = 1, L_2 = 1 + 2*x, L_3 = 1 + 3*x, L_4 = 1 + 4*x + 2*x^2, L_5 = 1 + 5*x + 5*x^2, L_6 = 1 + 6*x + 9*x^2 + 2*x^3, L_7 = 1 + 7*x + 14*x^2 + 7*x^3, L_8 = 1 + 8*x + 20*x^2 + 16*x^3 + 2*x^4, L_9 = 1 + 9*x + 27*x^2 + 30*x^3 + 9*x^4, ...
The other version (probably the more official one) begins L_0(x) = 2, L_1(x) = x, L_2(x) = 2 + x^2, L_3(x) = 3*x + x^3, L_4(x) = 2 + 4*x^2 + x^4, L_5(x) = 5*x + 5*x^3 + x^5, L_6(x) = 2 + 9*x^2 + 6*x^4 + x^6, L_7(x) = 7*x + 14*x^3 + 7*x^5 + x^7, L_8(x) = 2 + 16*x^2 + 20*x^4 + 8*x^6 + x^8, L_9(x) = 9*x + 30*x^3 + 27*x^5 + 9*x^7 + x^9.
From _John Blythe Dobson_, Oct 11 2007: (Start)
Triangle begins:
  2;
  1;
  1,  2;
  1,  3;
  1,  4,  2;
  1,  5,  5;
  1,  6,  9,   2;
  1,  7, 14,   7;
  1,  8, 20,  16,   2;
  1,  9, 27,  30,   9;
  1, 10, 35,  50,  25,   2;
  1, 11, 44,  77,  55,  11;
  1, 12, 54, 112, 105,  36,   2;
  1, 13, 65, 156, 182,  91,  13;
  1, 14, 77, 210, 294, 196,  49,  2;
  1, 15, 90, 275, 450, 378, 140, 15;
(End)
From _Peter Bala_, Mar 20 2025: (Start)
Let S = x + y and M = -x*y. Then the triangle gives the coefficients when expressing the symmetric polynomial x^n + y^n as a polynomial in S and M. For example,
x^2 + y^2 = S^2 + 2*M; x^3 + y^3 = S^3 + 3*S*M; x^4 + y^4 = S^4 + 4*(S^2)*M + 2*M^2;
x^5 + y^5 = S^5 + 5*(S^3)*M + 5*S*M^2; x^6 + y^6 = S^6 + 6*(S^4)*M + 9*(S^2)*M^2 + 2*M^3. See Woko. In general x^n + y^n = 2*(-i)^n *(sqrt(M))^n * T(n, i*S/(2*sqrt(M))), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. (End)

References

  • A. Brousseau, Fibonacci and Related Number Theoretic Tables. Fibonacci Association, San Jose, CA, 1972, p. 148.
  • C. D. Godsil, Algebraic Combinatorics, Chapman and Hall, New York, 1993.
  • Thomas Koshy, Fibonacci and Lucas Numbers with Applications. New York, etc.: John Wiley & Sons, 2001. (Chapter 13, "Pascal-like Triangles," is devoted to the present triangle.)
  • The Royal Society Newton Tercentenary Celebrations, Cambridge Univ. Press, 1947.

Links

Crossrefs

Cf. A029635, A061896, A111125, A113279, A114525, A127672, A127677, A156308, A217476, A263916.
Cf. A117179.

Programs

  • Maple
    T:= proc(n,k) if n=0 and k=0 then 2 elif k>floor(n/2) then 0 else n*binomial(n-k,k)/(n-k) fi end: for n from 0 to 15 do seq(T(n,k), k=0..floor(n/2)) od; # yields sequence in triangular form # Emeric Deutsch, Dec 25 2004
  • Mathematica
    t[0, 0] = 2; t[n_, k_] := Binomial[n-k, k] + Binomial[n-k-1, k-1]; Table[t[n, k], {n, 0, 16}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Dec 30 2013 *)
CoefficientList[Table[x^(n/2) LucasL[n, 1/Sqrt[x]], {n, 0, 15}], x] // Flatten (* Eric W. Weisstein, Apr 06 2017 *)
Table[Select[Reverse[CoefficientList[LucasL[n, x], x]], 0 < # &], {n, 0, 16}] // Flatten (* Robert G. Wilson v, May 03 2017 *)
CoefficientList[FunctionExpand @ Table[2 (-x)^(n/2) Cos[n ArcSec[2 Sqrt[-x]]], {n, 0, 15}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
CoefficientList[Table[2 (-x)^(n/2) ChebyshevT[n, 1/(2 Sqrt[-x])], {n, 0, 15}], x] // Flatten (* Eric W. Weisstein, Apr 03 2018 *)
  • PARI
    {T(n, k) = if( k<0 || 2*k>n, 0, binomial(n-k, k) + binomial(n-k-1, k-1) + (n==0))}; /* Michael Somos, Jul 15 2003 */

Formula

Row sums = A000032. T(2n, n-1) = A000290(n), T(2n+1, n-1) = A000330(n), T(2n, n-2) = A002415(n). T(n, k) = A029635(n-k, k), if n>0. - Michael Somos, Apr 02 1999
Lucas polynomial coefficients: 1, -n, n*(n-3)/2!, -n*(n-4)*(n-5)/3!, n*(n-5)*(n-6)*(n-7)/4!, - n*(n-6)*(n-7)*(n-8)*(n-9)/5!, ... - Herb Conn and Gary W. Adamson, May 28 2003
G.f.: (2-x)/(1-x-x^2*y). - Vladeta Jovovic, May 31 2003
T(n, k) = T(n-1, k) + T(n-2, k-1), n>1. T(n, 0) = 1, n>0. T(n, k) = binomial(n-k, k) + binomial(n-k-1, k-1) = n*binomial(n-k-1, k-1)/k, 0 <= 2*k <= n except T(0, 0) = 2. - Michael Somos, Apr 02 1999
T(n,k) = (n*(n-1-k)!)/(k!*(n-2*k)!), n>0, k>=0. - Alexander Elkins (alexander_elkins(AT)hotmail.com), Jun 09 2007
O.g.f.: 2-(2xt+1)xt/(-t+xt+(xt)^2). (Cf. A113279.) - Tom Copeland, Nov 07 2015
T(n,k) = A011973(n-1,k) + A011973(n-3,k-1) = A011973(n,k) - A011973(n-4,k-2) except for T(0,0)=T(2,1)=2. - Xiangyu Chen, Dec 24 2020
L_n(x) = ((x+sqrt(x^2+4))/2)^n + (-((x+sqrt(x^2+4))/2))^(-n). See metallic means. - William Krier, Sep 01 2023

Extensions

Improved description, more terms, etc., from Michael Somos

A029653 Numbers in (2,1)-Pascal triangle (by row).

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 2, 5, 4, 1, 2, 7, 9, 5, 1, 2, 9, 16, 14, 6, 1, 2, 11, 25, 30, 20, 7, 1, 2, 13, 36, 55, 50, 27, 8, 1, 2, 15, 49, 91, 105, 77, 35, 9, 1, 2, 17, 64, 140, 196, 182, 112, 44, 10, 1, 2, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 2, 21, 100, 285
Offset: 0

Views

Author

Mohammad K. Azarian

Keywords

Comments

Reverse of A029635. Row sums are A003945. Diagonal sums are Fibonacci(n+2) = Sum_{k=0..floor(n/2)} (2n-3k)*C(n-k,n-2k)/(n-k). - Paul Barry, Jan 30 2005
Riordan array ((1+x)/(1-x), x/(1-x)). The signed triangle (-1)^(n-k)T(n,k) or ((1-x)/(1+x), x/(1+x)) is the inverse of A055248. Row sums are A003945. Diagonal sums are F(n+2). - Paul Barry, Feb 03 2005
Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...) = (1, 3, 7, 15, 31, 63, 127, ...) - (0, 0, 1, 3, 7, 15, 31, ...); where (1, 3, 7, 15, ...) = A000225. - Gary W. Adamson, Apr 22 2007
Triangle T(n,k), read by rows, given by (2,-1,0,0,0,0,0,0,0,...) DELTA (1,0,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 17 2011
A029653 is jointly generated with A208510 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x)+1. See the Mathematica section. - Clark Kimberling, Feb 28 2012
For a closed-form formula for arbitrary left and right borders of Pascal like triangle, see A228196. - Boris Putievskiy, Aug 18 2013
For a closed-form formula for generalized Pascal's triangle, see A228576. - Boris Putievskiy, Sep 04 2013
The n-th row polynomial is (2 + x)*(1 + x)^(n-1) for n >= 1. More generally, the n-th row polynomial of the Riordan array ( (1-a*x)/(1-b*x), x/(1-b*x) ) is (b - a + x)*(b + x)^(n-1) for n >= 1. - Peter Bala, Feb 25 2018

Examples

			The triangle T(n,k) begins:
n\k 0  1  2   3   4   5   6   7  8  9 10 ...
0:  1
1:  2  1
2:  2  3  1
3:  2  5  4   1
4:  2  7  9   5   1
5:  2  9 16  14   6   1
6:  2 11 25  30  20   7   1
7:  2 13 36  55  50  27   8   1
8:  2 15 49  91 105  77  35   9  1
9:  2 17 64 140 196 182 112  44 10  1
10: 2 19 81 204 336 378 294 156 54 11  1
... Reformatted. - _Wolfdieter Lang_, Jan 09 2015
With the array M(k) as defined in the Formula section, the infinite product M(0)*M(1)*M(2)*... begins
/1        \/1         \/1        \      /1        \
|2 1      ||0 1       ||0 1      |      |2 1      |
|2 1 1    ||0 2 1     ||0 0 1    |... = |2 3 1    |
|2 1 1 1  ||0 2 1 1   ||0 0 2 1  |      |2 5 4 1  |
|2 1 1 1 1||0 2 1 1 1 ||0 0 2 1 1|      |2 7 9 5 1|
|...      ||...       ||...      |      |...      |
- _Peter Bala_, Dec 27 2014

References

  • Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

Crossrefs

(d, 1) Pascal triangles: A007318(d=1), A093560(3), A093561(4), A093562(5), A093563(6), A093564(7), A093565(8), A093644(9), A093645(10).
Cf. A003945, A208510, A228196, A228576.
Cf. A078812, A106195.

Programs

  • Haskell
    a029653 n k = a029653_tabl !! n !! k
a029653_row n = a029653_tabl !! n
a029653_tabl = [1] : iterate
               (\xs -> zipWith (+) ([0] ++ xs) (xs ++ [0])) [2, 1]
-- Reinhard Zumkeller, Dec 16 2013
  • Maple
    A029653 :=  proc(n,k)
if n = 0 then
  1;
else
  binomial(n-1, k)+binomial(n, k)
fi
end proc: # R. J. Mathar, Jun 30 2013
  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A029653 *)
(* Clark Kimberling, Feb 28 2012 *)
  • Python
    from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(v(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017
  • Python
    from math import comb, isqrt
def A029653(n): return comb(r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),a:=n-comb(r+1,2))*((r<<1)-a)//r if n else 1 # Chai Wah Wu, Nov 12 2024

Formula

T(n, k) = C(n-2, k-1) + C(n-2, k) + C(n-1, k-1) + C(n-1, k) except for n=0.
G.f.: (1 + x + y + xy)/(1 - y - xy). - Ralf Stephan, May 17 2004
T(n, k) = (2n-k)*binomial(n, n-k)/n, n, k > 0. - Paul Barry, Jan 30 2005
Sum_{k=0..n} T(n, k)*x^k gives A003945-A003954 for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - Philippe Deléham, Jul 10 2005
T(n, k) = C(n-1, k) + C(n, k). - Philippe Deléham, Jul 10 2005
Equals A097806 * A007318, i.e., the pairwise operator * Pascal's Triangle as infinite lower triangular matrices. - Gary W. Adamson, Apr 22 2007
From Peter Bala, Dec 27 2014: (Start)
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(2 + 5*x + 4*x^2/2! + x^3/3!) = 2 + 7*x + 16*x^2/2! + 30*x^3/3! + 50*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ).
Let M denote the lower unit triangular array with 1's on the main diagonal and 1's everywhere else below the main diagonal except for the first column which consists of the sequence [1,2,2,2,...]. For k = 0,1,2,... define M(k) to be the lower unit triangular block array
/I_k 0\
\ 0 M/ having the k X k identity matrix I_k as the upper left block; in particular, M(0) = M. Then the present triangle equals the infinite product M(0)*M(1)*M(2)*... (which is clearly well-defined). See the Example section. (End)

Extensions

More terms from James Sellers

A179070 a(1)=a(2)=a(3)=1, a(4)=3; thereafter a(n) = a(n-1) + a(n-3).

Original entry on oeis.org

1, 1, 1, 3, 4, 5, 8, 12, 17, 25, 37, 54, 79, 116, 170, 249, 365, 535, 784, 1149, 1684, 2468, 3617, 5301, 7769, 11386, 16687, 24456, 35842, 52529, 76985, 112827, 165356, 242341, 355168, 520524, 762865, 1118033, 1638557, 2401422, 3519455, 5158012, 7559434
Offset: 1

Views

Author

Mark Dols, Jun 27 2010

Keywords

Comments

Also (essentially), coordination sequence for (2,4,infinity) tiling of hyperbolic plane. - N. J. A. Sloane, Dec 29 2015
Column sums of shifted (1,2) Pascal array:
1 1 1 1 1 1 1 1 1
......2 3 4 5 6 7
............2 5 9
.................
----------------- +
1 1 1 3 4 5 8 ...
a(n+1) is the number of multus bitstrings of length n with no runs of 2 0's. - Steven Finch, Mar 25 2020
From Areebah Mahdia and Greg Dresden, Jun 13 2020: (Start)
For n >= 5, a(n) gives the number of ways to tile the following board of length n-3 with squares and trominos:
.
|||
|||_ _ _
|||_|||_|_| ... . (End)

Links

Crossrefs

Cf. A000930, A029635, A097333, A214626.
Coordination sequences for triangular tilings of hyperbolic space: A001630, A007283, A054886, A078042, A096231, A163876, A179070, A265057, A265058, A265059, A265060, A265061, A265062, A265063, A265064, A265065, A265066, A265067, A265068, A265069, A265070, A265071, A265072, A265073, A265074, A265075, A265076, A265077.

Programs

Formula

a(n) = A000930(n-1) + A000930(n-4).
G.f.: x - x^2*(1+2*x^2) / ( -1+x+x^3 ). - R. J. Mathar, Oct 30 2011
a(n) = A000930(n-2)+2*A000930(n-4) for n>3. - R. J. Mathar, May 19 2024

Extensions

Simpler definition from N. J. A. Sloane, Aug 29 2013

A051924 a(n) = binomial(2*n,n) - binomial(2*n-2,n-1); or (3n-2)*C(n-1), where C = Catalan numbers (A000108).

Original entry on oeis.org

1, 4, 14, 50, 182, 672, 2508, 9438, 35750, 136136, 520676, 1998724, 7696444, 29716000, 115000920, 445962870, 1732525830, 6741529080, 26270128500, 102501265020, 400411345620, 1565841089280, 6129331763880, 24014172955500, 94163002754652, 369507926510352
Offset: 1

Views

Author

Barry E. Williams, Dec 19 1999

Keywords

Comments

Number of partitions with Ferrers plots that fit inside an n X n box, but not in an n-1 X n-1 box. - Wouter Meeussen, Dec 10 2001
From Benoit Cloitre, Jan 29 2002: (Start)
Let m(1,j)=j, m(i,1)=i and m(i,j) = m(i-1,j) + m(i,j-1); then a(n) = m(n,n):
1 2 3 4 ...
2 4 7 11 ...
3 7 14 25 ...
4 11 25 50 ... (End)
This sequence also gives the number of clusters and non-crossing partitions of type D_n. - F. Chapoton, Jan 31 2005
If Y is a 2-subset of a 2n-set X then a(n) is the number of (n+1)-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
Prefaced with a 1: (1, 1, 4, 14, 50, ...) and convolved with the Catalan sequence = A097613: (1, 2, 7, 25, 91, ...). - Gary W. Adamson, May 15 2009
Total number of up steps before the second return in all Dyck n-paths. - David Scambler, Aug 21 2012
Conjecture: a(n) mod n^2 = n+2 iff n is an odd prime. - Gary Detlefs, Feb 19 2013
First differences of A000984 and A030662. - J. M. Bergot, Jun 22 2013
From R. J. Mathar, Jun 30 2013: (Start)
Equivalent to the Meeussen comment and the Bergot comment: The array view of A007318 is
1, 1, 1, 1, 1, 1,
1, 2, 3, 4, 5, 6,
1, 3, 6, 10, 15, 21,
1, 4, 10, 20, 35, 56,
1, 5, 15, 35, 70, 126,
1, 6, 21, 56, 126, 252,
and a(n) are the hook sums Sum_{k=0..n} A(n,k) + Sum_{r=0..n-1} A(r,n). (End)
From Gus Wiseman, Apr 12 2019: (Start)
Equivalent to Wouter Meeussen's comment, a(n) is the number of integer partitions (of any positive integer) such that the maximum of the length and the largest part is n. For example, the a(1) = 1 through a(3) = 14 partitions are:
(1) (2) (3)
(11) (31)
(21) (32)
(22) (33)
(111)
(211)
(221)
(222)
(311)
(321)
(322)
(331)
(332)
(333)
(End)
Coxeter-Catalan numbers for Coxeter groups of type D_n [Armstrong]. - N. J. A. Sloane, Mar 09 2022
a(n+1) is the number of ways that a best of n pairs contest with early termination can go. For example, the first stage of an association football (soccer) penalty-kick shoot out has n=5 pairs of shots and there are a(6)=672 distinct ways it can go. For n=2 pairs, writing G for goal and M for miss, and listing the up-to-four shots in chronological order with teams alternating shots, the n(3)=14 possibilities are MMMM, MMMG, MMGM, MMGG, MGM, MGGM, MGGG, GMMM, GMMG, GMG, GGMM, GGMG, GGGM, and GGGG. Not all four shots are taken in two cases because it becomes impossible for one team to overcome the lead of the other team. - Lee A. Newberg, Jul 20 2024

Examples

			Sums of {1}, {2, 1, 1}, {2, 2, 3, 3, 2, 1, 1}, {2, 2, 4, 5, 7, 6, 7, 5, 5, 3, 2, 1, 1}, ...

References

  • Drew Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274 16; See Table 2.8.

Links

Crossrefs

Left-central elements of the (1, 2)-Pascal triangle A029635.
Column sums of A096771.
Cf. A000108, A024482 (diagonal from 2), A076540 (diagonal from 3), A000124 (row from 2), A004006 (row from 3), A006522 (row from 4).
Cf. A128064; first differences of A000984.
Cf. A097613.
Cf. A115720, A252464, A257990, A263297, A325189, A325192, A325193.

Programs

  • Haskell
    a051924 n = a051924_list !! (n-1)
a051924_list = zipWith (-) (tail a000984_list) a000984_list
-- Reinhard Zumkeller, May 25 2013
  • Magma
    [Binomial(2*n, n)-Binomial(2*n-2, n-1): n in [1..28]]; // Vincenzo Librandi, Dec 21 2016
  • Maple
    C:= n-> binomial(2*n, n)/(n+1): seq((n+1)*C(n)-n*C(n-1), n=1..25); # Emeric Deutsch, Jan 08 2008
Z:=(1-z-sqrt(1-4*z))/sqrt(1-4*z): Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..24); # Zerinvary Lajos, Jan 01 2007
a := n -> 2^(-2+2*n)*GAMMA(-1/2+n)*(3*n-2)/(sqrt(Pi)*GAMMA(1+n)):
seq(simplify(a(n)), n=1..24); # Peter Luschny, Dec 14 2015
  • Mathematica
    Table[Binomial[2n,n]-Binomial[2n-2,n-1],{n,30}] (* Harvey P. Dale, Jan 15 2012 *)
  • PARI
    a(n)=binomial(2*n,n)-binomial(2*n-2,n-1) \\ Charles R Greathouse IV, Jun 25 2013
  • PARI
    {a(n)=polcoeff((1-x) / sqrt(1-4*x +x*O(x^n)) - 1,n)}
for(n=1,30,print1(a(n),", ")) \\ Paul D. Hanna, Nov 08 2014
  • PARI
    {a(n)=polcoeff( sum(m=1, n, x^m * sum(k=0, m, binomial(m, k)^2 * x^k) / (1-x +x*O(x^n))^(2*m)), n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 08 2014
  • Sage
    a = lambda n: 2^(-2+2*n)*gamma(n-1/2)*(3*n-2)/(sqrt(pi)*gamma(1+n))
[a(n) for n in (1..120)] # Peter Luschny, Dec 14 2015

Formula

G.f.: (1-x) / sqrt(1-4*x) - 1. - Paul D. Hanna, Nov 08 2014
G.f.: Sum_{n>=1} x^n/(1-x)^(2*n) * Sum_{k=0..n} C(n,k)^2 * x^k. - Paul D. Hanna, Nov 08 2014
a(n+1) = binomial(2*n, n) + 2*Sum_{i=0..n-1} binomial(n+i, i) (V's in Pascal's Triangle). - Jon Perry Apr 13 2004
a(n) = n*C(n-1) - (n-1)*C(n-2), where C(n) = A000108(n) = Catalan(n). For example, a(5) = 50 = 5*C(4) - 4*C(3) - 5*14 - 3*5 = 70 - 20. Triangle A128064 as an infinite lower triangular matrix * A000108 = A051924 prefaced with a 1: (1, 1, 4, 14, 50, 182, ...). - Gary W. Adamson, May 15 2009
Sum of 3 central terms of Pascal's triangle: 2*C(2+2*n, n)+C(2+2*n, 1+n). - Zerinvary Lajos, Dec 20 2005
a(n+1) = A051597(2n,n). - Philippe Deléham, Nov 26 2006
The sequence 1,1,4,... has a(n) = C(2*n,n)-C(2*(n-1),n-1) = 0^n+Sum_{k=0..n} C(n-1,k-1)*A002426(k), and g.f. given by (1-x)/(1-2*x-2*x^2/(1-2*x-x^2/(1-2*x-x^2/(1-2*x-x^2/(1-.... (continued fraction). - Paul Barry, Oct 17 2009
a(n) = (3*n-2)*(2*n-2)!/(n*(n-1)!^2) = A001700(n) + A001791(n-1). - David Scambler, Aug 21 2012
D-finite with recurrence: a(n) = 2*(3*n-2)*(2*n-3)*a(n-1)/(n*(3*n-5)). - Alois P. Heinz, Apr 25 2014
a(n) = 2^(-2+2*n)*Gamma(-1/2+n)*(3*n-2)/(sqrt(Pi)*Gamma(1+n)). - Peter Luschny, Dec 14 2015
a(n) ~ (3/4)*4^n*(1-(7/24)/n-(7/128)/n^2-(85/3072)/n^3-(581/32768)/n^4-(2611/262144)/n^5)/sqrt(n*Pi). - Peter Luschny, Dec 16 2015
E.g.f.: ((1 - x)*BesselI(0,2*x) + x*BesselI(1,2*x))*exp(2*x) - 1. - Ilya Gutkovskiy, Dec 20 2016
a(n) = 2 * A097613(n) for n > 1. - Bruce J. Nicholson, Jan 06 2019
Sum_{n>=1} a(n)/8^n = 7/(4*sqrt(2)) - 1. - Amiram Eldar, May 06 2023

Extensions

Edited by N. J. A. Sloane, May 03 2008, at the suggestion of R. J. Mathar
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