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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A000142 Factorial numbers: n! = 1*2*3*4*...*n (order of symmetric group S_n, number of permutations of n letters).

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000, 121645100408832000, 2432902008176640000, 51090942171709440000, 1124000727777607680000
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

The earliest publication that discusses this sequence appears to be the Sepher Yezirah [Book of Creation], circa AD 300. (See Knuth, also the Zeilberger link.) - N. J. A. Sloane, Apr 07 2014
For n >= 1, a(n) is the number of n X n (0,1) matrices with each row and column containing exactly one entry equal to 1.
This sequence is the BinomialMean transform of A000354. (See A075271 for definition.) - John W. Layman, Sep 12 2002 [This is easily verified from the Paul Barry formula for A000354, by interchanging summations and using the formula: Sum_k (-1)^k C(n-i, k) = KroneckerDelta(i,n). - David Callan, Aug 31 2003]
Number of distinct subsets of T(n-1) elements with 1 element A, 2 elements B, ..., n - 1 elements X (e.g., at n = 5, we consider the distinct subsets of ABBCCCDDDD and there are 5! = 120). - Jon Perry, Jun 12 2003
n! is the smallest number with that prime signature. E.g., 720 = 2^4 * 3^2 * 5. - Amarnath Murthy, Jul 01 2003
a(n) is the permanent of the n X n matrix M with M(i, j) = 1. - Philippe Deléham, Dec 15 2003
Given n objects of distinct sizes (e.g., areas, volumes) such that each object is sufficiently large to simultaneously contain all previous objects, then n! is the total number of essentially different arrangements using all n objects. Arbitrary levels of nesting of objects are permitted within arrangements. (This application of the sequence was inspired by considering leftover moving boxes.) If the restriction exists that each object is able or permitted to contain at most one smaller (but possibly nested) object at a time, the resulting sequence begins 1,2,5,15,52 (Bell Numbers?). Sets of nested wooden boxes or traditional nested Russian dolls come to mind here. - Rick L. Shepherd, Jan 14 2004
From Michael Somos, Mar 04 2004; edited by M. F. Hasler, Jan 02 2015: (Start)
Stirling transform of [2, 2, 6, 24, 120, ...] is A052856 = [2, 2, 4, 14, 76, ...].
Stirling transform of [1, 2, 6, 24, 120, ...] is A000670 = [1, 3, 13, 75, ...].
Stirling transform of [0, 2, 6, 24, 120, ...] is A052875 = [0, 2, 12, 74, ...].
Stirling transform of [1, 1, 2, 6, 24, 120, ...] is A000629 = [1, 2, 6, 26, ...].
Stirling transform of [0, 1, 2, 6, 24, 120, ...] is A002050 = [0, 1, 5, 25, 140, ...].
Stirling transform of (A165326*A089064)(1...) = [1, 0, 1, -1, 8, -26, 194, ...] is [1, 1, 2, 6, 24, 120, ...] (this sequence). (End)
First Eulerian transform of 1, 1, 1, 1, 1, 1... The first 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 first-order Eulerian number [A008292]. - Ross La Haye, Feb 13 2005
Conjecturally, 1, 6, and 120 are the only numbers which are both triangular and factorial. - Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Mar 30 2005
n! is the n-th finite difference of consecutive n-th powers. E.g., for n = 3, [0, 1, 8, 27, 64, ...] -> [1, 7, 19, 37, ...] -> [6, 12, 18, ...] -> [6, 6, ...]. - Bryan Jacobs (bryanjj(AT)gmail.com), Mar 31 2005
a(n+1) = (n+1)! = 1, 2, 6, ... has e.g.f. 1/(1-x)^2. - Paul Barry, Apr 22 2005
Write numbers 1 to n on a circle. Then a(n) = sum of the products of all n - 2 adjacent numbers. E.g., a(5) = 1*2*3 + 2*3*4 + 3*4*5 + 4*5*1 +5*1*2 = 120. - Amarnath Murthy, Jul 10 2005
The number of chains of maximal length in the power set of {1, 2, ..., n} ordered by the subset relation. - Rick L. Shepherd, Feb 05 2006
The number of circular permutations of n letters for n >= 0 is 1, 1, 1, 2, 6, 24, 120, 720, 5040, 40320, ... - Xavier Noria (fxn(AT)hashref.com), Jun 04 2006
a(n) is the number of deco polyominoes of height n (n >= 1; see definitions in the Barcucci et al. references). - Emeric Deutsch, Aug 07 2006
a(n) is the number of partition tableaux of size n. See Steingrimsson/Williams link for the definition. - David Callan, Oct 06 2006
Consider the n! permutations of the integer sequence [n] = 1, 2, ..., n. The i-th permutation consists of ncycle(i) permutation cycles. Then, if the Sum_{i=1..n!} 2^ncycle(i) runs from 1 to n!, we have Sum_{i=1..n!} 2^ncycle(i) = (n+1)!. E.g., for n = 3 we have ncycle(1) = 3, ncycle(2) = 2, ncycle(3) = 1, ncycle(4) = 2, ncycle(5) = 1, ncycle(6) = 2 and 2^3 + 2^2 + 2^1 + 2^2 + 2^1 + 2^2 = 8 + 4 + 2 + 4 + 2 + 4 = 24 = (n+1)!. - Thomas Wieder, Oct 11 2006
a(n) is the number of set partitions of {1, 2, ..., 2n - 1, 2n} into blocks of size 2 (perfect matchings) in which each block consists of one even and one odd integer. For example, a(3) = 6 counts 12-34-56, 12-36-45, 14-23-56, 14-25-36, 16-23-45, 16-25-34. - David Callan, Mar 30 2007
Consider the multiset M = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4, ...] = [1, 2, 2, ..., n x 'n'] and form the set U (where U is a set in the strict sense) of all subsets N (where N may be a multiset again) of M. Then the number of elements |U| of U is equal to (n+1)!. E.g. for M = [1, 2, 2] we get U = [[], [2], [2, 2], [1], [1, 2], [1, 2, 2]] and |U| = 3! = 6. This observation is a more formal version of the comment given already by Rick L. Shepherd, Jan 14 2004. - Thomas Wieder, Nov 27 2007
For n >= 1, a(n) = 1, 2, 6, 24, ... are the positions corresponding to the 1's in decimal expansion of Liouville's constant (A012245). - Paul Muljadi, Apr 15 2008
Triangle A144107 has n! for row sums (given n > 0) with right border n! and left border A003319, the INVERTi transform of (1, 2, 6, 24, ...). - Gary W. Adamson, Sep 11 2008
Equals INVERT transform of A052186 and row sums of triangle A144108. - Gary W. Adamson, Sep 11 2008
From Abdullahi Umar, Oct 12 2008: (Start)
a(n) is also the number of order-decreasing full transformations (of an n-chain).
a(n-1) is also the number of nilpotent order-decreasing full transformations (of an n-chain). (End)
n! is also the number of optimal broadcast schemes in the complete graph K_{n}, equivalent to the number of binomial trees embedded in K_{n} (see Calin D. Morosan, Information Processing Letters, 100 (2006), 188-193). - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
Let S_{n} denote the n-star graph. The S_{n} structure consists of n S_{n-1} structures. This sequence gives the number of edges between the vertices of any two specified S_{n+1} structures in S_{n+2} (n >= 1). - K.V.Iyer, Mar 18 2009
Chromatic invariant of the sun graph S_{n-2}.
It appears that a(n+1) is the inverse binomial transform of A000255. - Timothy Hopper, Aug 20 2009
a(n) is also the determinant of a square matrix, An, whose coefficients are the reciprocals of beta function: a{i, j} = 1/beta(i, j), det(An) = n!. - Enrique Pérez Herrero, Sep 21 2009
The asymptotic expansions of the exponential integrals E(x, m = 1, n = 1) ~ exp(-x)/x*(1 - 1/x + 2/x^2 - 6/x^3 + 24/x^4 + ...) and E(x, m = 1, n = 2) ~ exp(-x)/x*(1 - 2/x + 6/x^2 - 24/x^3 + ...) lead to the factorial numbers. See A163931 and A130534 for more information. - Johannes W. Meijer, Oct 20 2009
Satisfies A(x)/A(x^2), A(x) = A173280. - Gary W. Adamson, Feb 14 2010
a(n) = G^n where G is the geometric mean of the first n positive integers. - Jaroslav Krizek, May 28 2010
Increasing colored 1-2 trees with choice of two colors for the rightmost branch of nonleaves. - Wenjin Woan, May 23 2011
Number of necklaces with n labeled beads of 1 color. - Robert G. Wilson v, Sep 22 2011
The sequence 1!, (2!)!, ((3!)!)!, (((4!)!)!)!, ..., ((...(n!)!)...)! (n times) grows too rapidly to have its own entry. See Hofstadter.
The e.g.f. of 1/a(n) = 1/n! is BesselI(0, 2*sqrt(x)). See Abramowitz-Stegun, p. 375, 9.3.10. - Wolfdieter Lang, Jan 09 2012
a(n) is the length of the n-th row which is the sum of n-th row in triangle A170942. - Reinhard Zumkeller, Mar 29 2012
Number of permutations of elements 1, 2, ..., n + 1 with a fixed element belonging to a cycle of length r does not depend on r and equals a(n). - Vladimir Shevelev, May 12 2012
a(n) is the number of fixed points in all permutations of 1, ..., n: in all n! permutations, 1 is first exactly (n-1)! times, 2 is second exactly (n-1)! times, etc., giving (n-1)!*n = n!. - Jon Perry, Dec 20 2012
For n >= 1, a(n-1) is the binomial transform of A000757. See Moreno-Rivera. - Luis Manuel Rivera Martínez, Dec 09 2013
Each term is divisible by its digital root (A010888). - Ivan N. Ianakiev, Apr 14 2014
For m >= 3, a(m-2) is the number hp(m) of acyclic Hamiltonian paths in a simple graph with m vertices, which is complete except for one missing edge. For m < 3, hp(m)=0. - Stanislav Sykora, Jun 17 2014
a(n) is the number of increasing forests with n nodes. - Brad R. Jones, Dec 01 2014
The factorial numbers can be calculated by means of the recurrence n! = (floor(n/2)!)^2 * sf(n) where sf(n) are the swinging factorials A056040. This leads to an efficient algorithm if sf(n) is computed via prime factorization. For an exposition of this algorithm see the link below. - Peter Luschny, Nov 05 2016
Treeshelves are ordered (plane) binary (0-1-2) increasing trees where the nodes of outdegree 1 come in 2 colors. There are n! treeshelves of size n, and classical Françon's bijection maps bijectively treeshelves into permutations. - Sergey Kirgizov, Dec 26 2016
Satisfies Benford's law [Diaconis, 1977; Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017
a(n) = Sum((d_p)^2), where d_p is the number of standard tableaux in the Ferrers board of the integer partition p and summation is over all integer partitions p of n. Example: a(3) = 6. Indeed, the partitions of 3 are [3], [2,1], and [1,1,1], having 1, 2, and 1 standard tableaux, respectively; we have 1^2 + 2^2 + 1^2 = 6. - Emeric Deutsch, Aug 07 2017
a(n) is the n-th derivative of x^n. - Iain Fox, Nov 19 2017
a(n) is the number of maximum chains in the n-dimensional Boolean cube {0,1}^n in respect to the relation "precedes". It is defined as follows: for arbitrary vectors u, v of {0,1}^n, such that u = (u_1, u_2, ..., u_n) and v = (v_1, v_2, ..., v_n), "u precedes v" if u_i <= v_i, for i=1, 2, ..., n. - Valentin Bakoev, Nov 20 2017
a(n) is the number of shortest paths (for example, obtained by Breadth First Search) between the nodes (0,0,...,0) (i.e., the all-zeros vector) and (1,1,...,1) (i.e., the all-ones vector) in the graph H_n, corresponding to the n-dimensional Boolean cube {0,1}^n. The graph is defined as H_n = (V_n, E_n), where V_n is the set of all vectors of {0,1}^n, and E_n contains edges formed by each pair adjacent vectors. - Valentin Bakoev, Nov 20 2017
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = sigma(gcd(i,j)) for 1 <= i,j <= n. - Bernard Schott, Dec 05 2018
a(n) is also the number of inversion sequences of length n. A length n inversion sequence e_1, e_2, ..., e_n is a sequence of n integers such that 0 <= e_i < i. - Juan S. Auli, Oct 14 2019
The term "factorial" ("factorielle" in French) was coined by the French mathematician Louis François Antoine Arbogast (1759-1803) in 1800. The notation "!" was first used by the French mathematician Christian Kramp (1760-1826) in 1808. - Amiram Eldar, Apr 16 2021
Also the number of signotopes of rank 2, i.e., mappings X:{{1..n} choose 2}->{+,-} such that for any three indices a < b < c, the sequence X(a,b), X(a,c), X(b,c) changes its sign at most once (see Felsner-Weil reference). - Manfred Scheucher, Feb 09 2022
a(n) is also the number of labeled commutative semisimple rings with n elements. As an example the only commutative semisimple rings with 4 elements are F_4 and F_2 X F_2. They both have exactly 2 automorphisms, hence a(4)=24/2+24/2=24. - Paul Laubie, Mar 05 2024
a(n) is the number of extremely unlucky Stirling permutations of order n+1; i.e., the number of Stirling permutations of order n+1 that have exactly one lucky car. - Bridget Tenner, Apr 09 2024

Examples

			There are 3! = 1*2*3 = 6 ways to arrange 3 letters {a, b, c}, namely abc, acb, bac, bca, cab, cba.
Let n = 2. Consider permutations of {1, 2, 3}. Fix element 3. There are a(2) = 2 permutations in each of the following cases: (a) 3 belongs to a cycle of length 1 (permutations (1, 2, 3) and (2, 1, 3)); (b) 3 belongs to a cycle of length 2 (permutations (3, 2, 1) and (1, 3, 2)); (c) 3 belongs to a cycle of length 3 (permutations (2, 3, 1) and (3, 1, 2)). - _Vladimir Shevelev_, May 13 2012
G.f. = 1 + x + 2*x^2 + 6*x^3 + 24*x^4 + 120*x^5 + 720*x^6 + 5040*x^7 + ...

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. 833.
  • A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 125; also p. 90, ex. 3.
  • Florian Cajori, A History of Mathematical Notations, Dover edition (2012), pars. 448-449.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 64-66.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.1 Symbols Galore, p. 106.
  • Douglas R. Hofstadter, Fluid concepts & creative analogies: computer models of the fundamental mechanisms of thought, Basic Books, 1995, pages 44-46.
  • A. N. Khovanskii. The Application of Continued Fractions and Their Generalizations to Problem in Approximation Theory. Groningen: Noordhoff, Netherlands, 1963. See p. 141 (10.19).
  • D. E. Knuth, The Art of Computer Programming, Vol. 3, Section 5.1.2, p. 23. [From N. J. A. Sloane, Apr 07 2014]
  • J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 693 pp. 90, 297, Ellipses Paris 2004.
  • 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.
  • R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
  • Sepher Yezirah [Book of Creation], circa AD 300. See verse 52.
  • 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).
  • Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, pages 19-24.
  • D. Stanton and D. White, Constructive Combinatorics, Springer, 1986; see p. 91.
  • Carlo Suares, Sepher Yetsira, Shambhala Publications, 1976. See verse 52.
  • David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1987, pp. 102.

Links

Crossrefs

Cf. A000165, A001044, A001563, A003422, A009445, A010050, A012245, A033312, A034886, A038507, A047920, A048631.
Factorial base representation: A007623.
Cf. A003319, A052186, A144107, A144108. - Gary W. Adamson, Sep 11 2008
Complement of A063992. - Reinhard Zumkeller, Oct 11 2008
Cf. A053657, A163176. - Jonathan Sondow, Jul 26 2009
Cf. A173280. - Gary W. Adamson, Feb 14 2010
Boustrophedon transforms: A230960, A230961.
Cf. A233589.
Cf. A245334.
A row of the array in A249026.
Cf. A001013 (multiplicative closure).
For factorials with initial digit d (1 <= d <= 9) see A045509, A045510, A045511, A045516, A045517, A045518, A282021, A045519; A045520, A045521, A045522, A045523, A045524, A045525, A045526, A045527, A045528, A045529.
Cf. A000169, A075513, A152917, A258773.

Programs

  • Axiom
    [factorial(n) for n in 0..10]
  • GAP
    List([0..22],Factorial); # Muniru A Asiru, Dec 05 2018
  • Haskell
    a000142 :: (Enum a, Num a, Integral t) => t -> a
a000142 n = product [1 .. fromIntegral n]
a000142_list = 1 : zipWith (*) [1..] a000142_list
-- Reinhard Zumkeller, Mar 02 2014, Nov 02 2011, Apr 21 2011
  • Julia
    print([factorial(big(n)) for n in 0:28]) # Paul Muljadi, May 01 2024
  • Magma
    a:= func< n | Factorial(n) >; [ a(n) : n in [0..10]];
  • Maple
    A000142 := n -> n!; seq(n!,n=0..20);
spec := [ S, {S=Sequence(Z) }, labeled ]; seq(combstruct[count](spec,size=n), n=0..20);
# Maple program for computing cycle indices of symmetric groups
M:=6: f:=array(0..M): f[0]:=1: print(`n= `,0); print(f[0]); f[1]:=x[1]: print(`n= `, 1); print(f[1]); for n from 2 to M do f[n]:=expand((1/n)*add( x[l]*f[n-l],l=1..n)); print(`n= `, n); print(f[n]); od:
with(combstruct):ZL0:=[S,{S=Set(Cycle(Z,card>0))},labeled]: seq(count(ZL0,size=n),n=0..20); # Zerinvary Lajos, Sep 26 2007
  • Mathematica
    Table[Factorial[n], {n, 0, 20}] (* Stefan Steinerberger, Mar 30 2006 *)
FoldList[#1 #2 &, 1, Range@ 20] (* Robert G. Wilson v, May 07 2011 *)
Range[20]! (* Harvey P. Dale, Nov 19 2011 *)
RecurrenceTable[{a[n] == n*a[n - 1], a[0] == 1}, a, {n, 0, 22}] (* Ray Chandler, Jul 30 2015 *)
  • PARI
    a(n)=prod(i=1, n, i) \\ Felix Fröhlich, Aug 17 2014
  • PARI
    {a(n) = if(n<0, 0, n!)}; /* Michael Somos, Mar 04 2004 */
  • Python
    for i in range(1, 1000):
    y = i
    for j in range(1, i):
       y *= i - j
    print(y, "\n")
  • Python
    import math
for i in range(1, 1000):
    math.factorial(i)
    print("")
# Ruskin Harding, Feb 22 2013
  • Sage
    [factorial(n) for n in (1..22)] # Giuseppe Coppoletta, Dec 05 2014
  • Scala
    (1: BigInt).to(24: BigInt).scanLeft(1: BigInt)( * ) // Alonso del Arte, Mar 02 2019

Formula

Exp(x) = Sum_{m >= 0} x^m/m!. - Mohammad K. Azarian, Dec 28 2010
Sum_{i=0..n} (-1)^i * i^n * binomial(n, i) = (-1)^n * n!. - Yong Kong (ykong(AT)curagen.com), Dec 26 2000
Sum_{i=0..n} (-1)^i * (n-i)^n * binomial(n, i) = n!. - Peter C. Heinig (algorithms(AT)gmx.de), Apr 10 2007
The sequence trivially satisfies the recurrence a(n+1) = Sum_{k=0..n} binomial(n,k) * a(k)*a(n-k). - Robert FERREOL, Dec 05 2009
D-finite with recurrence: a(n) = n*a(n-1), n >= 1. n! ~ sqrt(2*Pi) * n^(n+1/2) / e^n (Stirling's approximation).
a(0) = 1, a(n) = subs(x = 1, (d^n/dx^n)(1/(2-x))), n = 1, 2, ... - Karol A. Penson, Nov 12 2001
E.g.f.: 1/(1-x). - Michael Somos, Mar 04 2004
a(n) = Sum_{k=0..n} (-1)^(n-k)*A000522(k)*binomial(n, k) = Sum_{k=0..n} (-1)^(n-k)*(x+k)^n*binomial(n, k). - Philippe Deléham, Jul 08 2004
Binomial transform of A000166. - Ross La Haye, Sep 21 2004
a(n) = Sum_{i=1..n} ((-1)^(i-1) * sum of 1..n taken n - i at a time) - e.g., 4! = (1*2*3 + 1*2*4 + 1*3*4 + 2*3*4) - (1*2 + 1*3 + 1*4 + 2*3 + 2*4 + 3*4) + (1 + 2 + 3 + 4) - 1 = (6 + 8 + 12 + 24) - (2 + 3 + 4 + 6 + 8 + 12) + 10 - 1 = 50 - 35 + 10 - 1 = 24. - Jon Perry, Nov 14 2005
a(n) = (n-1)*(a(n-1) + a(n-2)), n >= 2. - Matthew J. White, Feb 21 2006
1 / a(n) = determinant of matrix whose (i,j) entry is (i+j)!/(i!(j+1)!) for n > 0. This is a matrix with Catalan numbers on the diagonal. - Alexander Adamchuk, Jul 04 2006
Hankel transform of A074664. - Philippe Deléham, Jun 21 2007
For n >= 2, a(n-2) = (-1)^n*Sum_{j=0..n-1} (j+1)*Stirling1(n,j+1). - Milan Janjic, Dec 14 2008
From Paul Barry, Jan 15 2009: (Start)
G.f.: 1/(1-x-x^2/(1-3x-4x^2/(1-5x-9x^2/(1-7x-16x^2/(1-9x-25x^2... (continued fraction), hence Hankel transform is A055209.
G.f. of (n+1)! is 1/(1-2x-2x^2/(1-4x-6x^2/(1-6x-12x^2/(1-8x-20x^2... (continued fraction), hence Hankel transform is A059332. (End)
a(n) = Product_{p prime} p^(Sum_{k > 0} floor(n/p^k)) by Legendre's formula for the highest power of a prime dividing n!. - Jonathan Sondow, Jul 24 2009
a(n) = A053657(n)/A163176(n) for n > 0. - Jonathan Sondow, Jul 26 2009
It appears that a(n) = (1/0!) + (1/1!)*n + (3/2!)*n*(n-1) + (11/3!)*n*(n-1)*(n-2) + ... + (b(n)/n!)*n*(n-1)*...*2*1, where a(n) = (n+1)! and b(n) = A000255. - Timothy Hopper, Aug 12 2009
Sum_{n >= 0} 1/a(n) = e. - Jaume Oliver Lafont, Mar 03 2009
a(n) = a(n-1)^2/a(n-2) + a(n-1), n >= 2. - Jaume Oliver Lafont, Sep 21 2009
a(n) = Gamma(n+1). - Enrique Pérez Herrero, Sep 21 2009
a(n) = A173333(n,1). - Reinhard Zumkeller, Feb 19 2010
a(n) = A_{n}(1) where A_{n}(x) are the Eulerian polynomials. - Peter Luschny, Aug 03 2010
a(n) = n*(2*a(n-1) - (n-1)*a(n-2)), n > 1. - Gary Detlefs, Sep 16 2010
1/a(n) = -Sum_{k=1..n+1} (-2)^k*(n+k+2)*a(k)/(a(2*k+1)*a(n+1-k)). - Groux Roland, Dec 08 2010
From Vladimir Shevelev, Feb 21 2011: (Start)
a(n) = Product_{p prime, p <= n} p^(Sum_{i >= 1} floor(n/p^i)).
The infinitary analog of this formula is: a(n) = Product_{q terms of A050376 <= n} q^((n)_q), where (n)_q denotes the number of those numbers <= n for which q is an infinitary divisor (for the definition see comment in A037445). (End)
The terms are the denominators of the expansion of sinh(x) + cosh(x). - Arkadiusz Wesolowski, Feb 03 2012
G.f.: 1 / (1 - x / (1 - x / (1 - 2*x / (1 - 2*x / (1 - 3*x / (1 - 3*x / ... )))))). - Michael Somos, May 12 2012
G.f. 1 + x/(G(0)-x) where G(k) = 1 - (k+1)*x/(1 - x*(k+2)/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 14 2012
G.f.: W(1,1;-x)/(W(1,1;-x) - x*W(1,2;-x)), where W(a,b,x) = 1 - a*b*x/1! + a*(a+1)*b*(b+1)*x^2/2! - ... + a*(a+1)*...*(a+n-1)*b*(b+1)*...*(b+n-1)*x^n/n! + ...; see [A. N. Khovanskii, p. 141 (10.19)]. - Sergei N. Gladkovskii, Aug 15 2012
From Sergei N. Gladkovskii, Dec 26 2012: (Start)
G.f.: A(x) = 1 + x/(G(0) - x) where G(k) = 1 + (k+1)*x - x*(k+2)/G(k+1); (continued fraction).
Let B(x) be the g.f. for A051296, then A(x) = 2 - 1/B(x). (End)
G.f.: 1 + x*(G(0) - 1)/(x-1) where G(k) = 1 - (2*k+1)/(1-x/(x - 1/(1 - (2*k+2)/(1-x/(x - 1/G(k+1) ))))); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: 1 + x*(1 - G(0))/(sqrt(x)-x) where G(k) = 1 - (k+1)*sqrt(x)/(1-sqrt(x)/(sqrt(x)-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 25 2013
G.f.: 1 + x/G(0) where G(k) = 1 - x*(k+2)/( 1 - x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
a(n) = det(S(i+1, j), 1 <= i, j <=n ), where S(n,k) are Stirling numbers of the second kind. - Mircea Merca, Apr 04 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: G(0), where G(k) = 1 + x*(2*k+1)/(1 - x*(2*k+2)/(x*(2*k+2) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 07 2013
a(n) = P(n-1, floor(n/2)) * floor(n/2)! * (n - (n-2)*((n+1) mod 2)), where P(n, k) are the k-permutations of n objects, n > 0. - Wesley Ivan Hurt, Jun 07 2013
a(n) = a(n-2)*(n-1)^2 + a(n-1), n > 1. - Ivan N. Ianakiev, Jun 18 2013
a(n) = a(n-2)*(n^2-1) - a(n-1), n > 1. - Ivan N. Ianakiev, Jun 30 2013
G.f.: 1 + x/Q(0), m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 24 2013
a(n) = A245334(n,n). - Reinhard Zumkeller, Aug 31 2014
a(n) = Product_{i = 1..n} A014963^floor(n/i) = Product_{i = 1..n} A003418(floor(n/i)). - Matthew Vandermast, Dec 22 2014
a(n) = round(Sum_{k>=1} log(k)^n/k^2), for n>=1, which is related to the n-th derivative of the Riemann zeta function at x=2 as follows: round((-1)^n * zeta^(n)(2)). Also see A073002. - Richard R. Forberg, Dec 30 2014
a(n) ~ Sum_{j>=0} j^n/e^j, where e = A001113. When substituting a generic variable for "e" this infinite sum is related to Eulerian polynomials. See A008292. This approximation of n! is within 0.4% at n = 2. See A255169. Accuracy, as a percentage, improves rapidly for larger n. - Richard R. Forberg, Mar 07 2015
a(n) = Product_{k=1..n} (C(n+1, 2)-C(k, 2))/(2*k-1); see Masanori Ando link. - Michel Marcus, Apr 17 2015
Sum_{n>=0} a(n)/(a(n + 1)*a(n + 2)) = Sum_{n>=0} 1/((n + 2)*(n + 1)^2*a(n)) = 2 - exp(1) - gamma + Ei(1) = 0.5996203229953..., where gamma = A001620, Ei(1) = A091725. - Ilya Gutkovskiy, Nov 01 2016
a(2^n) = 2^(2^n - 1) * 1!! * 3!! * 7!! * ... * (2^n - 1)!!. For example, 16! = 2^15*(1*3)*(1*3*5*7)*(1*3*5*7*9*11*13*15) = 20922789888000. - Peter Bala, Nov 01 2016
a(n) = sum(prod(B)), where the sum is over all subsets B of {1,2,...,n-1} and where prod(B) denotes the product of all the elements of set B. If B is a singleton set with element b, then we define prod(B)=b, and, if B is the empty set, we define prod(B) to be 1. For example, a(4)=(1*2*3)+(1*2)+(1*3)+(2*3)+(1)+(2)+(3)+1=24. - Dennis P. Walsh, Oct 23 2017
Sum_{n >= 0} 1/(a(n)*(n+2)) = 1. - Multiplying the denominator by (n+2) in Jaume Oliver Lafont's entry above creates a telescoping sum. - Fred Daniel Kline, Nov 08 2020
O.g.f.: Sum_{k >= 0} k!*x^k = Sum_{k >= 0} (k+y)^k*x^k/(1 + (k+y)*x)^(k+1) for arbitrary y. - Peter Bala, Mar 21 2022
E.g.f.: 1/(1 + LambertW(-x*exp(-x))) = 1/(1-x), see A258773. -(1/x)*substitute(z = x*exp(-x), z*(d/dz)LambertW(-z)) = 1/(1 - x). See A075513. Proof: Use the compositional inverse (x*exp(-x))^[-1] = -LambertW(-z). See A000169 or A152917, and Richard P. Stanley: Enumerative Combinatorics, vol. 2, p. 37, eq. (5.52). - Wolfdieter Lang, Oct 17 2022
Sum_{k >= 1} 1/10^a(k) = A012245 (Liouville constant). - Bernard Schott, Dec 18 2022
From David Ulgenes, Sep 19 2023: (Start)
1/a(n) = (e/(2*Pi*n)*Integral_{x=-oo..oo} cos(x-n*arctan(x))/(1+x^2)^(n/2) dx). Proof: take the real component of Laplace's integral for 1/Gamma(x).
a(n) = Integral_{x=0..1} e^(-t)*LerchPhi(1/e, -n, t) dt. Proof: use the relationship Gamma(x+1) = Sum_{n >= 0} Integral_{t=n..n+1} e^(-t)t^x dt = Sum_{n >= 0} Integral_{t=0..1} e^(-(t+n))(t+n)^x dt and interchange the order of summation and integration.
Conjecture: a(n) = 1/(2*Pi)*Integral_{x=-oo..oo}(n+i*x+1)!/(i*x+1)-(n+i*x-1)!/(i*x-1)dx. (End)
a(n) = floor(b(n)^n / (floor(((2^b(n) + 1) / 2^n)^b(n)) mod 2^b(n))), where b(n) = (n + 1)^(n + 2) = A007778(n+1). Joint work with Mihai Prunescu. - Lorenzo Sauras Altuzarra, Oct 18 2023
a(n) = e^(Integral_{x=1..n+1} Psi(x) dx) where Psi(x) is the digamma function. - Andrea Pinos, Jan 10 2024
a(n) = Integral_{x=0..oo} e^(-x^(1/n)) dx, for n > 0. - Ridouane Oudra, Apr 20 2024
O.g.f.: N(x) = hypergeometric([1,1], [], x) = LaplaceTransform(x/(1-x))/x, satisfying x^2*N'(x) + (x-1)*N(x) + 1 = 0, with N(0) = 1. - Wolfdieter Lang, May 31 2025

A040000 a(0)=1; a(n)=2 for n >= 1.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2
Offset: 0

Views

Author

N. J. A. Sloane, Dec 11 1999

Keywords

Comments

Continued fraction expansion of sqrt(2) is 1 + 1/(2 + 1/(2 + 1/(2 + ...))).
Inverse binomial transform of Mersenne numbers A000225(n+1) = 2^(n+1) - 1. - Paul Barry, Feb 28 2003
A Chebyshev transform of 2^n: if A(x) is the g.f. of a sequence, map it to ((1-x^2)/(1+x^2))A(x/(1+x^2)). - Paul Barry, Oct 31 2004
An inverse Catalan transform of A068875 under the mapping g(x)->g(x(1-x)). A068875 can be retrieved using the mapping g(x)->g(xc(x)), where c(x) is the g.f. of A000108. A040000 and A068875 may be described as a Catalan pair. - Paul Barry, Nov 14 2004
Sequence of electron arrangement in the 1s 2s and 3s atomic subshells. Cf. A001105, A016825. - Jeremy Gardiner, Dec 19 2004
Binomial transform of A165326. - Philippe Deléham, Sep 16 2009
Let m=2. We observe that a(n) = Sum_{k=0..floor(n/2)} binomial(m,n-2*k). Then there is a link with A113311 and A115291: it is the same formula with respectively m=3 and m=4. We can generalize this result with the sequence whose g.f. is given by (1+z)^(m-1)/(1-z). - Richard Choulet, Dec 08 2009
With offset 1: number of permutations where |p(i) - p(i+1)| <= 1 for n=1,2,...,n-1. This is the identical permutation and (for n>1) its reversal.
Equals INVERT transform of bar(1, 1, -1, -1, ...).
Eventual period is (2). - Zak Seidov, Mar 05 2011
Also decimal expansion of 11/90. - Vincenzo Librandi, Sep 24 2011
a(n) = 3 - A054977(n); right edge of the triangle in A182579. - Reinhard Zumkeller, May 07 2012
With offset 1: minimum cardinality of the range of a periodic sequence with (least) period n. Of course the range's maximum cardinality for a purely periodic sequence with (least) period n is n. - Rick L. Shepherd, Dec 08 2014
With offset 1: n*a(1) + (n-1)*a(2) + ... + 2*a(n-1) + a(n) = n^2. - Warren Breslow, Dec 12 2014
With offset 1: decimal expansion of gamma(4) = 11/9 where gamma(n) = Cp(n)/Cv(n) is the n-th Poisson's constant. For the definition of Cp and Cv see A272002. - Natan Arie Consigli, Sep 11 2016
a(n) equals the number of binary sequences of length n where no two consecutive terms differ. Also equals the number of binary sequences of length n where no two consecutive terms are the same. - David Nacin, May 31 2017
a(n) is the period of the continued fractions for sqrt((n+2)/(n+1)) and sqrt((n+1)/(n+2)). - A.H.M. Smeets, Dec 05 2017
Also, number of self-avoiding walks and coordination sequence for the one-dimensional lattice Z. - Sean A. Irvine, Jul 27 2020

Examples

			sqrt(2) = 1.41421356237309504... = 1 + 1/(2 + 1/(2 + 1/(2 + 1/(2 + ...)))). - _Harry J. Smith_, Apr 21 2009
G.f. = 1 + 2*x + 2*x^2 + 2*x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 2*x^8 + ...
11/90 = 0.1222222222222222222... - _Natan Arie Consigli_, Sep 11 2016

References

  • A. Beiser, Concepts of Modern Physics, 2nd Ed., McGraw-Hill, 1973.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
  • Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §4.4 Powers and Roots, p. 144.
  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 276-278.

Links

Crossrefs

Convolution square is A008574.
See A003945 etc. for (1+x)/(1-k*x).
From Jaume Oliver Lafont, Mar 26 2009: (Start)
Sum_{0<=k<=n} a(k) = A005408(n).
Prod_{0<=k<=n} a(k) = A000079(n). (End)
Cf. A113311, A115291, A171418, A171440, A171441, A171442, A171443.
Cf. A000674 (boustrophedon transform).
Cf. A001333/A000129 (continued fraction convergents).
Cf. A000122, A002193 (sqrt(2) decimal expansion), A006487 (Egyptian fraction).
Cf. Other continued fractions for sqrt(a^2+1) = (a, 2a, 2a, 2a....): A040002 (contfrac(sqrt(5)) = (2,4,4,...)), A040006, A040012, A040020, A040030, A040042, A040056, A040072, A040090, A040110 (contfrac(sqrt(122)) = (11,22,22,...)), A040132, A040156, A040182, A040210, A040240, A040272, A040306, A040342, A040380, A040420 (contfrac(sqrt(442)) = (21,42,42,...)), A040462, A040506, A040552, A040600, A040650, A040702, A040756, A040812, A040870, A040930 (contfrac(sqrt(962)) = (31,62,62,...)).

Programs

  • Haskell
    a040000 0 = 1; a040000 n = 2
a040000_list = 1 : repeat 2  -- Reinhard Zumkeller, May 07 2012
  • Maple
    Digits := 100: convert(evalf(sqrt(2)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[2],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
a[ n_] := 2 - Boole[n == 0]; (* Michael Somos, Dec 28 2014 *)
PadRight[{1},120,2] (* or *) RealDigits[11/90, 10, 120][[1]] (* Harvey P. Dale, Jul 12 2025 *)
  • PARI
    {a(n) = 2-!n}; /* Michael Somos, Apr 16 2007 */
  • PARI
    a(n)=1+sign(n)  \\ Jaume Oliver Lafont, Mar 26 2009
  • PARI
    allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(2)); for (n=0, 20000, write("b040000.txt", n, " ", x[n+1]));  \\ Harry J. Smith, Apr 21 2009

Formula

G.f.: (1+x)/(1-x). - Paul Barry, Feb 28 2003
a(n) = 2 - 0^n; a(n) = Sum_{k=0..n} binomial(1, k). - Paul Barry, Oct 16 2004
a(n) = n*Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k, k)*2^(n-2*k)/(n-k). - Paul Barry, Oct 31 2004
A040000(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A068875(n-k). - Paul Barry, Nov 14 2004
From Michael Somos, Apr 16 2007: (Start)
Euler transform of length 2 sequence [2, -1].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u-v)*(u+v) - 2*v*(u-w).
E.g.f.: 2*exp(x) - 1.
a(n) = a(-n) for all n in Z (one possible extension to n<0). (End)
G.f.: (1-x^2)/(1-x)^2. - Jaume Oliver Lafont, Mar 26 2009
G.f.: exp(2*atanh(x)). - Jaume Oliver Lafont, Oct 20 2009
a(n) = Sum_{k=0..n} A108561(n,k)*(-1)^k. - Philippe Deléham, Nov 17 2013
a(n) = 1 + sign(n). - Wesley Ivan Hurt, Apr 16 2014
10 * 11/90 = 11/9 = (11/2 R)/(9/2 R) = Cp(4)/Cv(4) = A272005/A272004, with R = A081822 (or A070064). - Natan Arie Consigli, Sep 11 2016
a(n) = A001227(A000040(n+1)). - Omar E. Pol, Feb 28 2018

A112555 Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I) and consequently the matrix logarithm satisfies log(T) = T - I, where I is the identity matrix.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 1, 1, 1, 1, -1, -2, -2, 0, 1, 1, 3, 4, 2, 1, 1, -1, -4, -7, -6, -3, 0, 1, 1, 5, 11, 13, 9, 3, 1, 1, -1, -6, -16, -24, -22, -12, -4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, -1, -8, -29, -62, -86, -80, -50, -20, -5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, -1, -10, -46, -128, -239, -314, -296, -200, -95, -30, -6, 0
Offset: 0

Views

Author

Paul D. Hanna, Sep 21 2005

Keywords

Comments

Signed version of A108561. Row sums equal A084247. The n-th unsigned row sum = A001045(n) + 1 (Jacobsthal numbers). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms.
Equals row reversal of triangle A112468 up to sign, where A112468 is the Riordan array (1/(1-x),x/(1+x)). - Paul D. Hanna, Jan 20 2006
The elements here match A108561 in absolute value, but the signs are crucial to the properties that the matrix A112555 exhibits; the main property being T^m = I + m*(T - I). This property is not satisfied by A108561. - Paul D. Hanna, Nov 10 2009
Eigensequence of the triangle = A140165. - Gary W. Adamson, Jan 30 2009
Triangle T(n,k), read by rows, given by [1,-2,0,0,0,0,0,0,0,...] DELTA [1,0,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Sep 17 2009

Examples

			Triangle T begins:
   1;
   1,   1;
  -1,   0,   1;
   1,   1,   1,   1;
  -1,  -2,  -2,   0,   1;
   1,   3,   4,   2,   1,   1;
  -1,  -4,  -7,  -6,  -3,   0,   1;
   1,   5,  11,  13,   9,   3,   1,   1;
  -1,  -6, -16, -24, -22, -12,  -4,   0,   1;
   1,   7,  22,  40,  46,  34,  16,   4,   1,   1;
  -1,  -8, -29, -62, -86, -80, -50, -20,  -5,   0,   1;
  ...
Matrix log, log(T) = T - I, begins:
   0;
   1,  0;
  -1,  0,  0;
   1,  1,  1,  0;
  -1, -2, -2,  0,  0;
   1,  3,  4,  2,  1,  0;
  -1, -4, -7, -6, -3,  0,  0;
  ...
Matrix inverse, T^-1 = 2*I - T, begins:
   1;
  -1,  1;
   1,  0,  1;
  -1, -1, -1,  1;
   1,  2,  2,  0,  1;
  -1, -3, -4, -2, -1,  1;
  ...
where adjacent sums in row n of T^-1 gives row n+1 of T.

Links

Crossrefs

Cf. A112468 (reversed rows), A108561, A084247, A001045, A072547, A112556, A279006, A140165.
From Philippe Deléham, Oct 07 2009: (Start)
Sum_{k=0..n} T(n, k)*x^(n-k) = A165760(n), A165759(n), A165758(n), A165755(n), A165752(n), A165746(n), A165751(n), A165747(n), A000007(n), A000012(n), A084247(n), A165553(n), A165622(n), A165625(n), A165638(n), A165639(n), A165748(n), A165749(n), A165750(n) for x= -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively.
Sum_{k=0..n} T(n, k)*x^k = A166157(n), A166153(n), A166152(n), A166149(n), A166036(n), A166035(n), A091004(n+1), A077925(n), A000007(n), A165326(n), A084247(n), A165405(n), A165458(n), A165470(n), A165491(n), A165505(n), A165506(n), A165510(n), A165511(n) for x = -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. (End)

Programs

  • Mathematica
    Clear[t]; t[0, 0] = 1; t[n_, 0] = (-1)^(Mod[n, 2]+1); t[n_, n_] = 1; t[n_, k_] /; k == n-1 := t[n, k] = Mod[n, 2]; t[n_, k_] /; 0 < k < n-1 := t[n, k] = -t[n-1, k] - t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *)
  • PARI
    {T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff( polcoeff( (1+2*x+x*y)/((1-x*y)*(1+x+x*y)),n,X),k,Y)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))
  • PARI
    {T(n,k)=local(m=1,x=X+X*O(X^n),y=Y+Y*O(Y^k)); polcoeff(polcoeff(1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)),n,X),k,Y)}
for(n=0,12, for(k=0,n, print1(T(n,k),", "));print(""))
  • Sage
    def A112555_row(n):
    @cached_function
    def prec(n, k):
        if k==n: return 1
        if k==0: return 0
        return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
    return [(-1)^(n-k+1)*prec(n+1, k) for k in (1..n+1)]
for n in (0..12): print(A112555_row(n)) # Peter Luschny, Mar 16 2016

Formula

G.f.: 1/(1-x*y) + x/((1-x*y)*(1+x+x*y)).
The m-th matrix power T^m has the g.f.: 1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)).
Recurrence: T(n, k) = [T^-1](n-1, k) + [T^-1](n-1, k-1), where T^-1 is the matrix inverse of T.
From Peter Bala, Jun 23 2025: (Start)
T^z = exp(z*log(T)) = I + z*(T - I) for arbitrary complex z, where I is the identity array.
exp(T) = e*T. More generally, exp(z * T^u) = exp(z)*T^(u*z) = exp(z)*I + u*z*exp(z)*(T - I).
sin(z * T^u) = sin(z)*I + u*z*cos(z)*(T - I).
cos(z * T^u) = cos(z)*I - u*z*sin(z)*(T - I).
tan(z * T^u) = tan(z)*I + u*z*sec(z)^2*(T - I).
Chebyshev_T(n, T^u) = I + (n^2)*u*(T - I) and
Legendre_P(n, T^u) = I + (n*(n+1)/2)*u*(T - I).
More generally, for n >= 1,
Chebyshev_T(n, z*T^u) = Chebyshev_T(n, z)*I + n*u*z*Chebyshev_U(n-1, z)*(T - I) and
Legendre_P(n, z*T^u) = Legendre_P(n, z)*I + u*Q(n, z)*(T - I), where Q(1, z) = z and Q(n, z) = n*Legendre_P(n, z) + Q(n-1, z)/z for n > 1.
All the above properties may also hold for the triangle A279006. (End)

A105809 Riordan array (1/(1 - x - x^2), x/(1 - x)).

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 5, 7, 7, 4, 1, 8, 12, 14, 11, 5, 1, 13, 20, 26, 25, 16, 6, 1, 21, 33, 46, 51, 41, 22, 7, 1, 34, 54, 79, 97, 92, 63, 29, 8, 1, 55, 88, 133, 176, 189, 155, 92, 37, 9, 1, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10, 1, 144, 232, 364, 530, 674, 709, 591
Offset: 0

Views

Author

Paul Barry, May 04 2005

Keywords

Comments

Previous name was: A Fibonacci-Pascal matrix.
From Wolfdieter Lang, Oct 04 2014: (Start)
In the column k of this triangle (without leading zeros) is the k-fold iterated partial sums of the Fibonacci numbers, starting with 1. A000045(n+1), A000071(n+3), A001924(n+1), A014162(n+1), A014166(n+1), ..., n >= 0. See the Riordan property.
For a combinatorial interpretation of these iterated partial sums see the H. Belbachir and A. Belkhir link. There table 1 shows in the rows these columns. In their notation (with r = k) f^(k)(n) = T(k, n+k).
The A-sequence of this Riordan triangle is [1, 1] (see the recurrence for T(n, k), k >= 1, given in the formula section). The Z-sequence is A165326 = [1, repeat(1, -1)]. See the W. Lang link under A006232 for Riordan A- and Z-sequences. (End)

Examples

			The triangle T(n,k) begins:
n\k   0   1   2    3    4    5    6    7    8   9  10 11 12 13 ...
0:    1
1:    1   1
2:    2   2   1
3:    3   4   3    1
4:    5   7   7    4    1
5:    8  12  14   11    5    1
6:   13  20  26   25   16    6    1
7:   21  33  46   51   41   22    7    1
8:   34  54  79   97   92   63   29    8    1
9:   55  88 133  176  189  155   92   37    9   1
10:  89 143 221  309  365  344  247  129   46  10   1
11: 144 232 364  530  674  709  591  376  175  56  11  1
12: 233 376 596  894 1204 1383 1300  967  551 231  67 12  1
13: 377 609 972 1490 2098 2587 2683 2267 1518 782 298 79 13  1
... reformatted and extended - _Wolfdieter Lang_, Oct 03 2014
------------------------------------------------------------------
Recurrence from Z-sequence (see a comment above): 8 = T(0,5) = (+1)*5 + (+1)*7 + (-1)*7 + (+1)*4 + (-1)*1 = 8. - _Wolfdieter Lang_, Oct 04 2014

Links

Crossrefs

Cf. A165326 (Z-sequence), A027934 (row sums), A010049(n+1) (antidiagonal sums), A212804 (alternating row sums), inverse is A105810.
Some other Fibonacci-Pascal triangles: A027926, A036355, A037027, A074829, A109906, A111006, A114197, A162741, A228074.

Programs

  • Haskell
    a105809 n k = a105809_tabl !! n !! k
a105809_row n = a105809_tabl !! n
a105809_tabl = map fst $ iterate
   (\(u:_, vs) -> (vs, zipWith (+) ([u] ++ vs) (vs ++ [0]))) ([1], [1,1])
-- Reinhard Zumkeller, Aug 15 2013
  • Maple
    T := (n,k) -> `if`(n=0,1,binomial(n,k)*hypergeom([1,k/2-n/2,k/2-n/2+1/2], [k+1,-n], -4)); for n from 0 to 13 do seq(simplify(T(n,k)),k=0..n) od; # Peter Luschny, Oct 10 2014
  • Mathematica
    T[n_, k_] := Sum[Binomial[n-j, k+j], {j, 0, n}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] (* Jean-François Alcover, Jun 11 2019 *)

Formula

Riordan array (1/(1-x-x^2), x/(1-x)).
Triangle T(n, k) = Sum_{j=0..n} binomial(n-j, k+j); T(n, 0) = A000045(n+1);
T(n, m) = T(n-1, m-1) + T(n-1, m).
T(n, k) = Sum_{j=0..n} binomial(j, n+k-j). - Paul Barry, Oct 23 2006
G.f. of row polynomials Sum_{k=0..n} T(n, k)*x^k is (1 - z)/((1 - z - z^2)*(1 - (1 + x)*z)) (Riordan property). - Wolfdieter Lang, Oct 04 2014
T(n, k) = binomial(n, k)*hypergeom([1, k/2 - n/2, k/2 - n/2 + 1/2],[k + 1, -n], -4) for n > 0. - Peter Luschny, Oct 10 2014
From Wolfdieter Lang, Feb 13 2025: (Start)
Array A(k, n) = Sum_{j=0..n} F(j+1)*binomial(k-1+n-j, k-1), k >= 0, n >= 0, with F = A000045, (from Riordan triangle k-th convolution in columns without leading 0s).
A(k, n) = F(n+1+2*k) - Sum_{j=0..k-1} F(2*(k-j)-1) * binomial(n+1+j, j), (from iteration of partial sums).
Triangle T(n, k) = A(k, n-k) = Sum_{j=k..n} F(n-j+1) * binomial(j-1, k-1), 0 <= k <= n.
T(n, k) = F(n+1+k) - Sum_{j=0..k-1} F(2*(k-j)-1) * binomial(n - (k-1-j), j). (End)
T(n, k) = A027926(n, n+k), for 0 <= k <= n. - Wolfdieter Lang, Mar 08 2025

Extensions

Use first formula as a more descriptive name, Joerg Arndt, Jun 08 2021

A118827 2-adic continued fraction of zero, where a(n) = 1 if n is odd, otherwise -2*A006519(n/2).

Original entry on oeis.org

1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -16, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -32, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -16, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -64, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -16, 1, -2, 1, -4, 1, -2, 1, -8, 1, -2, 1, -4, 1, -2, 1, -32, 1, -2, 1
Offset: 1

Views

Author

Paul D. Hanna, May 01 2006

Keywords

Comments

Limit of convergents equals zero; only the 6th convergent is indeterminate. Other 2-adic continued fractions of zero are: A118821, A118824, A118830. A006519(n) is the highest power of 2 dividing n; A080277 = partial sums of A038712, where A038712(n) = 2*A006519(n) - 1.
Multiplicative because both A006519 and A165326 are. - Andrew Howroyd, Aug 01 2018

Examples

			For n >= 1, convergents A118828(k)/A118829(k):
  at k = 4*n: -1/(2*A080277(n));
  at k = 4*n+1: -1/(2*A080277(n)-1);
  at k = 4*n+2: -1/(2*A080277(n)-2);
  at k = 4*n-1: 0.
Convergents begin:
  1/1, -1/-2, 0/-1, -1/2, -1/1, 1/0, 0/1, 1/-8,
  1/-7, -1/6, 0/-1, -1/10, -1/9, 1/-8, 0/1, 1/-24,
  1/-23, -1/22, 0/-1, -1/26, -1/25, 1/-24, 0/1, 1/-32,
  1/-31, -1/30, 0/-1, -1/34, -1/33, 1/-32, 0/1, 1/-64, ...

Links

Crossrefs

Cf. A001620, A006519, A080277; convergents: A118828/A118829; variants: A118821, A118824, A118830; A100338, A165326.

Programs

  • Mathematica
    Array[If[OddQ@ #, 1, -2*2^(IntegerExponent[#, 2] - 1)] &, 99] (* Michael De Vlieger, Nov 06 2018 *)
  • PARI
    a(n)=local(p=+1,q=-2);if(n%2==1,p,q*2^valuation(n/2,2))

Formula

a(n) = A165326(n) * A006519(n). - Andrew Howroyd, Aug 01 2018
From Amiram Eldar, Oct 28 2023: (Start)
Multiplicative with a(2^e) = -2^e, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 - 2^(1-s) + 1/(2-2^s)).
Sum_{k=1..n} a(k) ~ (-1/(2*log(2))) * n *(log(n) + gamma - log(2)/2 - 1), where gamma is Euler's constant (A001620). (End)

A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

Original entry on oeis.org

1, 1, 0, 5, 1, 0, 61, 28, 1, 0, 1385, 1011, 123, 1, 0, 50521, 50666, 11706, 506, 1, 0, 2702765, 3448901, 1212146, 118546, 2041, 1, 0, 199360981, 308869464, 147485535, 24226000, 1130235, 8184, 1, 0
Offset: 0

Views

Author

Peter Luschny, Sep 20 2017

Keywords

Comments

The generalized Eulerian polynomials F_{m}(x) are defined F_{m; 0}(x) = 1 for all m >= 0 and for n > 0:
F_{0; n}(x) = Sum_{k=0..n} A097805(n, k)*(x-1)^(n-k) with coeffs. in A129186.
F_{1; n}(x) = Sum_{k=0..n} A131689(n, k)*(x-1)^(n-k) with coeffs. in A173018.
F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) with coeffs. in A292604.
F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) with coeffs. in A292605.
F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) with coeffs. in A292606.
The case m = 1 are the Eulerian polynomials whose coefficients are the Eulerian numbers which are displayed in Euler's triangle A173018.
Evaluated at x in {-1, 1, 0} these families of polynomials give for the first few m:
F_{m} : F_{0} F_{1} F_{2} F_{3} F_{4}
x = -1: A165326 A155585 A002105 A292609 A292607
x = 1: A000012 A000142 A000680 A014606 A014608 ... (m*n)!/m!^n
x = 0: -- A000012 A000364 A002115 A211212 ... m-alternating permutations of length m*n.
Note that the constant terms of the polynomials are the generalized Euler numbers as defined in A181985. In this sense generalized Euler numbers are also generalized Eulerian numbers.

Examples

			Triangle starts:
[n\k][    0        1        2       3     4  5  6]
--------------------------------------------------
[0][      1]
[1][      1,       0]
[2][      5,       1,       0]
[3][     61,      28,       1,      0]
[4][   1385,    1011,     123,      1,    0]
[5][  50521,   50666,   11706,    506,    1, 0]
[6][2702765, 3448901, 1212146, 118546, 2041, 1, 0]

References

  • G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.

Crossrefs

F_{0} = A129186, F_{1} = A173018, F_{2} is this triangle, F_{3} = A292605, F_{4} = A292606.
First column: A000364. Row sums: A000680. Alternating row sums: A002105.
Cf. A181985, A241171.

Programs

  • Maple
    Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292604_row := proc(n) if n = 0 then return [1] fi;
add(A241171(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292604_row(n) od;
  • Mathematica
    T[n_, k_] /; 1 <= k <= n := T[n, k] = k (2 k - 1) T[n - 1, k - 1] + k^2 T[n - 1, k]; T[, 1] = 1; T[, _] = 0;
F[2, 0][] = 1; F[2, n][x_] := Sum[T[n, k] (x - 1)^(n - k), {k, 0, n}];
row[n_] := If[n == 0, {1}, Append[CoefficientList[ F[2, n][x], x], 0]];
Table[row[n], {n, 0, 7}] (* Jean-François Alcover, Jul 06 2019 *)
  • Sage
    def A292604_row(n):
    if n == 0: return [1]
    S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..6): print(A292604_row(n))

Formula

F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.

A115164 a(n) = 3*a(n-1) + 4*a(n-2), with a(0) = 3, a(1) = 7, a(3) = 9, for n > 2.

Original entry on oeis.org

3, 7, 9, 55, 201, 823, 3273, 13111, 52425, 209719, 838857, 3355447, 13421769, 53687095, 214748361, 858993463, 3435973833, 13743895351, 54975581385, 219902325559, 879609302217, 3518437208887, 14073748835529, 56294995342135
Offset: 0

Views

Author

Roger L. Bagula, Mar 06 2006

Keywords

Links

Crossrefs

Cf. A115113, A115335.

Programs

  • Magma
    [3] cat [(4^(1+n) -19*(-1)^n)/5: n in [1..50]]; // G. C. Greubel, Nov 23 2018
  • Mathematica
    Join[{3}, LinearRecurrence[{3, 4}, {7, 9}, 50]]
  • Maxima
    (a[0] : 3, a[1] : 7, a[2] : 9, a[n] := 3*a[n-1] + 4*a[n-2], makelist(a[n], n, 0, 50)); /* Franck Maminirina Ramaharo, Nov 23 2018 */
  • PARI
    vector(50, n, n--; if(n==0, 3, (4^(1+n) -19*(-1)^n)/5)) \\ G. C. Greubel, Nov 23 2018
  • Sage
    [3] + [(4^(1+n) -19*(-1)^n)/5 for n in (1..50)] # G. C. Greubel, Nov 23 2018

Formula

From Colin Barker, Oct 31 2012: (Start)
a(n) = (4^(1 + n) - 19*(-1)^n)/5 for n > 0.
a(n) = 3*a(n-1) + 4*a(n-2) for n > 2.
G.f.: (24*x^2 + 2*x - 3)/((x + 1)*(4*x - 1)). (End)
From Franck Maminirina Ramaharo, Nov 23 2018: (Start)
a(n) = A115113(n) + A165326(n).
E.g.f.: (30 - 19*exp(-x) + 4*exp(4*x))/5. (End)

Extensions

Edited, and new name from Franck Maminirina Ramaharo, Nov 23 2018, after Colin Barker

A193730 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x) = (2x+1)^n and q(n,x) = (2x+1)^n.

Original entry on oeis.org

1, 2, 1, 4, 8, 3, 8, 28, 30, 9, 16, 80, 144, 108, 27, 32, 208, 528, 648, 378, 81, 64, 512, 1680, 2880, 2700, 1296, 243, 128, 1216, 4896, 10800, 14040, 10692, 4374, 729, 256, 2816, 13440, 36288, 60480, 63504, 40824, 14580, 2187, 512, 6400, 35328, 112896, 229824, 308448, 272160, 151632, 48114, 6561
Offset: 0

Views

Author

Clark Kimberling, Aug 04 2011

Keywords

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.
Triangle T(n,k), read by rows, given by (2,0,0,0,0,0,0,0,...) DELTA (1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 05 2011

Examples

			First six rows:
   1;
   2,   1;
   4,   8,   3;
   8,  28,  30,   9;
  16,  80, 144, 108,  27;
  32, 208, 528, 648, 378, 81;

Links

Crossrefs

Cf. A000079, A005053, A084938, A130129, A133494, A165326.
Cf. A193722, A193731, A199923.

Programs

  • Magma
    function T(n, k) // T = A193730
  if k lt 0 or k gt n then return 0;
  elif n lt 2 then return n-k+1;
  else return 2*T(n-1, k) + 3*T(n-1, k-1);
  end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2023
  • Mathematica
    (* First program *)
z = 8; a = 2; b = 1; c = 2; d = 1;
p[n_, x_] := (a*x + b)^n ; q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]     (* A193730 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]     (* A193731 *)
(* Second program *)
T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n<2, n-k+1, 2*T[n-1, k] + 3*T[n-1, k-1]]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2023 *)
  • SageMath
    def T(n, k): # T = A193730
    if (k<0 or k>n): return 0
    elif (n<2): return n-k+1
    else: return 2*T(n-1, k) + 3*T(n-1, k-1)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Nov 20 2023

Formula

T(n,k) = 3*T(n-1,k-1) + 2*T(n-1,k) with T(0,0)=T(1,1)=1 and T(1,0)=2. - Philippe Deléham, Oct 05 2011
G.f.: (1-2*x*y)/(1-2*x-3*x*y). - R. J. Mathar, Aug 11 2015
From G. C. Greubel, Nov 20 2023: (Start)
T(n, 0) = A000079(n).
T(n, 1) = A130129(n-1).
T(n, n) = A133494(n).
T(n, n-1) = A199923(n).
Sum_{k=0..n} T(n, k) = A005053(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A165326(n). (End)

A347171 Triangle read by rows where T(n,k) is the sum of Golay-Rudin-Shapiro terms GRS(j) (A020985) for j in the range 0 <= j < 2^n and having binary weight wt(j) = A000120(j) = k.

Original entry on oeis.org

1, 1, 1, 1, 2, -1, 1, 3, -1, 1, 1, 4, 0, 0, -1, 1, 5, 2, -2, 1, 1, 1, 6, 5, -4, 3, -2, -1, 1, 7, 9, -5, 3, -3, 3, 1, 1, 8, 14, -4, 0, 0, 2, -4, -1, 1, 9, 20, 0, -6, 6, -4, 0, 5, 1, 1, 10, 27, 8, -14, 12, -10, 8, -3, -6, -1, 1, 11, 35, 21, -22, 14, -10, 10, -11, 7, 7, 1
Offset: 0

Views

Author

Kevin Ryde, Aug 21 2021

Keywords

Comments

Doche and Mendès France form polynomials P_n(y) = Sum_{j=0..2^n-1} GRS(j) * y^wt(j) and here row n is the coefficients of P_n starting from the constant term, so P_n(y) = Sum_{k=0..n} T(n,k)*y^k. They conjecture that the number of real roots of P_n is A285869(n).
Row sum n is the sum of GRS terms from j = 0 to 2^n-1 inclusive, which Brillhart and Morton (Beispiel 6 page 129) show is A020986(2^n-1) = 2^ceiling(n/2) = A060546(n). The same follows by substituting y=1 in the P_n recurrence or the generating function.

Examples

			Triangle begins
        k=0 k=1 k=2 k=3 k=4 k=5 k=6 k=7
  n=0:   1
  n=1:   1,  1
  n=2:   1,  2, -1
  n=3:   1,  3, -1,  1
  n=4:   1,  4,  0,  0, -1
  n=5:   1,  5,  2, -2,  1,  1
  n=6:   1,  6,  5, -4,  3, -2, -1
  n=7:   1,  7,  9, -5,  3, -3,  3,  1
For T(5,3), those j in the range 0 <= j < 2^5 with wt(j) = 3 are
  j      =  7 11 13 14 19 21 22 25 26 28
  GRS(j) = +1 -1 -1 +1 -1 +1 -1 -1 -1 +1 total -2 = T(5,3)

Links

Crossrefs

Cf. A020985 (GRS), A020986 (GRS partial sums), A000120 (binary weight), A285869.
Columns k=0..3: A000012, A001477, A000096, A275874.
Cf. A165326 (main diagonal), A248157 (second diagonal negated).
Cf. A060546 (row sums), A104969 (row sums squared terms).
Cf. A329301 (antidiagonal sums).
Cf. A104967 (rows reversed, up to signs).

Programs

  • PARI
    my(M=Mod('x, 'x^2-(1-'y)*'x-2*'y)); row(n) = Vecrev(subst(lift(M^n),'x,'y+1));

Formula

T(n,k) = T(n-1,k) - T(n-1,k-1) + 2*T(n-2,k-1) for n>=2, and taking T(n,k)=0 if k<0 or k>n.
T(n,k) = (-1)^k * A104967(n,n-k).
Row polynomial P_n(y) = (1-y)*P_{n-1}(y) + 2*y*P_{n-2}(y) for n>=2. [Doche and Mendès France]
G.f.: (1 + 2*x*y)/(1 + x*(y-1) - 2*x^2*y).
Column g.f.: C_k(x) = 1/(1-x) for k=0 and C_k(x) = x^k * (2*x-1)^(k-1) / (1-x)^(k+1) for k>=1.
Showing 1-9 of 9 results.

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