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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A131978 A049614(n)/A131685(n).

Original entry on oeis.org

1, 1, 1, 4, 4, 24, 24, 192, 1728, 17280, 17280, 207360, 207360, 414720, 6220800, 99532800, 696729600, 12541132800, 12541132800, 250822656000, 5267275776000, 10534551552000, 10534551552000, 252829237248000, 1264146186240000, 12641461862400000, 3754514173132800000
Offset: 1

Views

Author

Alexander R. Povolotsky, Sep 21 2007

Keywords

Comments

Conjectured to be always integral.
It appears that every term > 4 is divisible by 3 - Alexander R. Povolotsky, Oct 18 2007

A000027 The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

For some authors, the terms "natural numbers" and "counting numbers" include 0, i.e., refer to the nonnegative integers A001477; the term "whole numbers" frequently also designates the whole set of (signed) integers A001057.
a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378).
Inverse Euler transform of A000219.
The rectangular array having A000027 as antidiagonals is the dispersion of the complement of the triangular numbers, A000217 (which triangularly form column 1 of this array). The array is also the transpose of A038722. - Clark Kimberling, Apr 05 2003
For nonzero x, define f(n) = floor(nx) - floor(n/x). Then f=A000027 if and only if x=tau or x=-tau. - Clark Kimberling, Jan 09 2005
Numbers of form (2^i)*k for odd k (i.e., n = A006519(n)*A000265(n)); thus n corresponds uniquely to an ordered pair (i,k) where i=A007814, k=A000265 (with A007814(2n)=A001511(n), A007814(2n+1)=0). - Lekraj Beedassy, Apr 22 2006
If the offset were changed to 0, we would have the following pattern: a(n)=binomial(n,0) + binomial(n,1) for the present sequence (number of regions in 1-space defined by n points), A000124 (number of regions in 2-space defined by n straight lines), A000125 (number of regions in 3-space defined by n planes), A000127 (number of regions in 4-space defined by n hyperplanes), A006261, A008859, A008860, A008861, A008862 and A008863, where the last six sequences are interpreted analogously and in each "... by n ..." clause an offset of 0 has been assumed, resulting in a(0)=1 for all of them, which corresponds to the case of not cutting with a hyperplane at all and therefore having one region. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
Define a number of points on a straight line to be in general arrangement when no two points coincide. Then these are the numbers of regions defined by n points in general arrangement on a straight line, when an offset of 0 is assumed. For instance, a(0)=1, since using no point at all leaves one region. The sequence satisfies the recursion a(n) = a(n-1) + 1. This has the following geometrical interpretation: Suppose there are already n-1 points in general arrangement, thus defining the maximal number of regions on a straight line obtainable by n-1 points, and now one more point is added in general arrangement. Then it will coincide with no other point and act as a dividing wall thereby creating one new region in addition to the a(n-1)=(n-1)+1=n regions already there, hence a(n)=a(n-1)+1. Cf. the comments on A000124 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
The sequence a(n)=n (for n=1,2,3) and a(n)=n+1 (for n=4,5,...) gives to the rank (minimal cardinality of a generating set) for the semigroup I_n\S_n, where I_n and S_n denote the symmetric inverse semigroup and symmetric group on [n]. - James East, May 03 2007
The sequence a(n)=n (for n=1,2), a(n)=n+1 (for n=3) and a(n)=n+2 (for n=4,5,...) gives the rank (minimal cardinality of a generating set) for the semigroup PT_n\T_n, where PT_n and T_n denote the partial transformation semigroup and transformation semigroup on [n]. - James East, May 03 2007
"God made the integers; all else is the work of man." This famous quotation is a translation of "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk," spoken by Leopold Kronecker in a lecture at the Berliner Naturforscher-Versammlung in 1886. Possibly the first publication of the statement is in Heinrich Weber's "Leopold Kronecker," Jahresberichte D.M.V. 2 (1893) 5-31. - Clark Kimberling, Jul 07 2007
Binomial transform of A019590, inverse binomial transform of A001792. - Philippe Deléham, Oct 24 2007
Writing A000027 as N, perhaps the simplest one-to-one correspondence between N X N and N is this: f(m,n) = ((m+n)^2 - m - 3n + 2)/2. Its inverse is given by I(k)=(g,h), where g = k - J(J-1)/2, h = J + 1 - g, J = floor((1 + sqrt(8k - 7))/2). Thus I(1)=(1,1), I(2)=(1,2), I(3)=(2,1) and so on; the mapping I fills the first-quadrant lattice by successive antidiagonals. - Clark Kimberling, Sep 11 2008
a(n) is also the mean of the first n odd integers. - Ian Kent, Dec 23 2008
Equals INVERTi transform of A001906, the even-indexed Fibonacci numbers starting (1, 3, 8, 21, 55, ...). - Gary W. Adamson, Jun 05 2009
These are also the 2-rough numbers: positive integers that have no prime factors less than 2. - Michael B. Porter, Oct 08 2009
Totally multiplicative sequence with a(p) = p for prime p. Totally multiplicative sequence with a(p) = a(p-1) + 1 for prime p. - Jaroslav Krizek, Oct 18 2009
Triangle T(k,j) of natural numbers, read by rows, with T(k,j) = binomial(k,2) + j = (k^2-k)/2 + j where 1 <= j <= k. In other words, a(n) = n = binomial(k,2) + j where k is the largest integer such that binomial(k,2) < n and j = n - binomial(k,2). For example, T(4,1)=7, T(4,2)=8, T(4,3)=9, and T(4,4)=10. Note that T(n,n)=A000217(n), the n-th triangular number. - Dennis P. Walsh, Nov 19 2009
Hofstadter-Conway-like sequence (see A004001): a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = 1, a(2) = 2. - Jaroslav Krizek, Dec 11 2009
a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2). - Leonid Bedratyuk, Jan 04 2010
Floyd's triangle read by rows. - Paul Muljadi, Jan 25 2010
Number of numbers between k and 2k where k is an integer. - Giovanni Teofilatto, Mar 26 2010
Generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1), r = 2; in an infinite set, row 2 of the array shown in A178568. - Gary W. Adamson, May 29 2010
1/n = continued fraction [n]. Let barover[n] = [n,n,n,...] = 1/k. Then k - 1/k = n. Example: [2,2,2,...] = (sqrt(2) - 1) = 1/k, with k = (sqrt(2) + 1). Then 2 = k - 1/k. - Gary W. Adamson, Jul 15 2010
Number of n-digit numbers the binary expansion of which contains one run of 1's. - Vladimir Shevelev, Jul 30 2010
From Clark Kimberling, Jan 29 2011: (Start)
Let T denote the "natural number array A000027":
1 2 4 7 ...
3 5 8 12 ...
6 9 13 18 ...
10 14 19 25 ...
T(n,k) = n+(n+k-2)*(n+k-1)/2. See A185787 for a list of sequences based on T, such as rows, columns, diagonals, and sub-arrays. (End)
The Stern polynomial B(n,x) evaluated at x=2. See A125184. - T. D. Noe, Feb 28 2011
The denominator in the Maclaurin series of log(2), which is 1 - 1/2 + 1/3 - 1/4 + .... - Mohammad K. Azarian, Oct 13 2011
As a function of Bernoulli numbers B_n (cf. A027641: (1, -1/2, 1/6, 0, -1/30, 0, 1/42, ...)): let V = a variant of B_n changing the (-1/2) to (1/2). Then triangle A074909 (the beheaded Pascal's triangle) * [1, 1/2, 1/6, 0, -1/30, ...] = the vector [1, 2, 3, 4, 5, ...]. - Gary W. Adamson, Mar 05 2012
Number of partitions of 2n+1 into exactly two parts. - Wesley Ivan Hurt, Jul 15 2013
Integers n dividing u(n) = 2u(n-1) - u(n-2); u(0)=0, u(1)=1 (Lucas sequence A001477). - Thomas M. Bridge, Nov 03 2013
For this sequence, the generalized continued fraction a(1)+a(1)/(a(2)+a(2)/(a(3)+a(3)/(a(4)+...))), evaluates to 1/(e-2) = A194807. - Stanislav Sykora, Jan 20 2014
Engel expansion of e-1 (A091131 = 1.71828...). - Jaroslav Krizek, Jan 23 2014
a(n) is the number of permutations of length n simultaneously avoiding 213, 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
a(n) is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - Manda Riehl, Aug 07 2014
a(n) = least k such that 2*Pi - Sum_{h=1..k} 1/(h^2 - h + 3/16) < 1/n. - Clark Kimberling, Sep 28 2014
a(n) = least k such that Pi^2/6 - Sum_{h=1..k} 1/h^2 < 1/n. - Clark Kimberling, Oct 02 2014
Determinants of the spiral knots S(2,k,(1)). a(k) = det(S(2,k,(1))). These knots are also the torus knots T(2,k). - Ryan Stees, Dec 15 2014
As a function, the restriction of the identity map on the nonnegative integers {0,1,2,3...}, A001477, to the positive integers {1,2,3,...}. - M. F. Hasler, Jan 18 2015
See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0: For k=1, A131685(k)=1, which implies that this is a well defined integer sequence. - Alexander R. Povolotsky, Apr 24 2015
a(n) is the number of compositions of n+2 into n parts avoiding the part 2. - Milan Janjic, Jan 07 2016
Does not satisfy Benford's law [Berger-Hill, 2017] - N. J. A. Sloane, Feb 07 2017
Parametrization for the finite multisubsets of the positive integers, where, for p_j the j-th prime, n = Product_{j} p_j^(e_j) corresponds to the multiset containing e_j copies of j ('Heinz encoding' -- see A056239, A003963, A289506, A289507, A289508, A289509). - Christopher J. Smyth, Jul 31 2017
The arithmetic function v_1(n,1) as defined in A289197. - Robert Price, Aug 22 2017
For n >= 3, a(n)=n is the least area that can be obtained for an irregular octagon drawn in a square of n units side, whose sides are parallel to the axes, with 4 vertices that coincide with the 4 vertices of the square, and the 4 remaining vertices having integer coordinates. See Affaire de Logique link. - Michel Marcus, Apr 28 2018
a(n+1) is the order of rowmotion on a poset defined by a disjoint union of chains of length n. - Nick Mayers, Jun 08 2018
Number of 1's in n-th generation of 1-D Cellular Automata using Rules 50, 58, 114, 122, 178, 186, 206, 220, 238, 242, 250 or 252 in the Wolfram numbering scheme, started with a single 1. - Frank Hollstein, Mar 25 2019
(1, 2, 3, 4, 5, ...) is the fourth INVERT transform of (1, -2, 3, -4, 5, ...). - Gary W. Adamson, Jul 15 2019

References

  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 1.
  • T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25.
  • John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 22.
  • W. Fulton and J. Harris, Representation theory: a first course, (1991), page 149. [From Leonid Bedratyuk, Jan 04 2010]
  • I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
  • R. E. Schwartz, You Can Count on Monsters: The First 100 numbers and Their Characters, A. K. Peters and MAA, 2010.
  • 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).

Links

Crossrefs

A001477 = nonnegative numbers.
Partial sums of A000012.
Cf. A001478, A001906, A007931, A007932, A027641, A074909, A089353 (multisets), A178568, A194807.
Cf. A026081 = integers in reverse alphabetical order in U.S. English, A107322 = English name for number and its reverse have the same number of letters, A119796 = zero through ten in alphabetical order of English reverse spelling, A005589, etc. Cf. A185787 (includes a list of sequences based on the natural number array A000027).
Cf. Boustrophedon transforms: A000737, A231179;
Cf. A038722 (mirrored when seen as triangle), A056011 (boustrophedon).
Cf. A048993, A048994, A000110 (see the Feb 03 2015 formula).
Cf. A289187, A000010, A008683, A000203, A049310.

Programs

Formula

a(2k+1) = A005408(k), k >= 0, a(2k) = A005843(k), k >= 1.
Multiplicative with a(p^e) = p^e. - David W. Wilson, Aug 01 2001
Another g.f.: Sum_{n>0} phi(n)*x^n/(1-x^n) (Apostol).
When seen as an array: T(k, n) = n+1 + (k+n)*(k+n+1)/2. Main diagonal is 2n*(n+1)+1 (A001844), antidiagonal sums are n*(n^2+1)/2 (A006003). - Ralf Stephan, Oct 17 2004
Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005
G.f.: x/(1-x)^2. E.g.f.: x*exp(x). a(n)=n. a(-n)=-a(n).
Series reversion of g.f. A(x) is x*C(-x)^2 where C(x) is the g.f. of A000108. - Michael Somos, Sep 04 2006
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 4*u*v. - Michael Somos, Oct 03 2006
Convolution of A000012 (the all-ones sequence) with itself. - Tanya Khovanova, Jun 22 2007
a(n) = 2*a(n-1)-a(n-2); a(1)=1, a(2)=2. a(n) = 1+a(n-1). - Philippe Deléham, Nov 03 2008
a(n) = A000720(A000040(n)). - Juri-Stepan Gerasimov, Nov 29 2009
a(n+1) = Sum_{k=0..n} A101950(n,k). - Philippe Deléham, Feb 10 2012
a(n) = Sum_{d | n} phi(d) = Sum_{d | n} A000010(d). - Jaroslav Krizek, Apr 20 2012
G.f.: x * Product_{j>=0} (1+x^(2^j))^2 = x * (1+2*x+x^2) * (1+2*x^2+x^4) * (1+2*x^4+x^8) * ... = x + 2x^2 + 3x^3 + ... . - Gary W. Adamson, Jun 26 2012
a(n) = det(binomial(i+1,j), 1 <= i,j <= n). - Mircea Merca, Apr 06 2013
E.g.f.: x*E(0), where E(k) = 1 + 1/(x - x^3/(x^2 + (k+1)/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 03 2013
From Wolfdieter Lang, Oct 09 2013: (Start)
a(n) = Product_{k=1..n-1} 2*sin(Pi*k/n), n > 1.
a(n) = Product_{k=1..n-1} (2*sin(Pi*k/(2*n)))^2, n > 1.
These identities are used in the calculation of products of ratios of lengths of certain lines in a regular n-gon. For the first identity see the Gradstein-Ryshik reference, p. 62, 1.392 1., bringing the first factor there to the left hand side and taking the limit x -> 0 (L'Hôpital). The second line follows from the first one. Thanks to Seppo Mustonen who led me to consider n-gon lengths products. (End)
a(n) = Sum_{j=0..k} (-1)^(j-1)*j*binomial(n,j)*binomial(n-1+k-j,k-j), k>=0. - Mircea Merca, Jan 25 2014
a(n) = A052410(n)^A052409(n). - Reinhard Zumkeller, Apr 06 2014
a(n) = Sum_{k=1..n^2+2*n} 1/(sqrt(k)+sqrt(k+1)). - Pierre CAMI, Apr 25 2014
a(n) = floor(1/sin(1/n)) = floor(cot(1/(n+1))) = ceiling(cot(1/n)). - Clark Kimberling, Oct 08 2014
a(n) = floor(1/(log(n+1)-log(n))). - Thomas Ordowski, Oct 10 2014
a(k) = det(S(2,k,1)). - Ryan Stees, Dec 15 2014
a(n) = 1/(1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 + ...). - Pierre CAMI, Jan 22 2015
a(n) = Sum_{m=0..n-1} Stirling1(n-1,m)*Bell(m+1), for n >= 1. This corresponds to Bell(m+1) = Sum_{k=0..m} Stirling2(m, k)*(k+1), for m >= 0, from the fact that Stirling2*Stirling1 = identity matrix. See A048993, A048994 and A000110. - Wolfdieter Lang, Feb 03 2015
a(n) = Sum_{k=1..2n-1}(-1)^(k+1)*k*(2n-k). In addition, surprisingly, a(n) = Sum_{k=1..2n-1}(-1)^(k+1)*k^2*(2n-k)^2. - Charlie Marion, Jan 05 2016
G.f.: x/(1-x)^2 = (x * r(x) *r(x^3) * r(x^9) * r(x^27) * ...), where r(x) = (1 + x + x^2)^2 = (1 + 2x + 3x^2 + 2x^3 + x^4). - Gary W. Adamson, Jan 11 2017
a(n) = floor(1/(Pi/2-arctan(n))). - Clark Kimberling, Mar 11 2020
a(n) = Sum_{d|n} mu(n/d)*sigma(d). - Ridouane Oudra, Oct 03 2020
a(n) = Sum_{k=1..n} phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 09 2021
a(n) = S(n-1, 2), with the Chebyshev S-polynomials A049310. - Wolfdieter Lang, Mar 09 2023
From Peter Bala, Nov 02 2024: (Start)
For positive integer m, a(n) = (1/m)* Sum_{k = 1..2*m*n-1} (-1)^(k+1) * k * (2*m*n - k) = (1/m) * Sum_{k = 1..2*m*n-1} (-1)^(k+1) * k^2 * (2*m*n - k)^2 (the case m = 1 is given above).
a(n) = Sum_{k = 0..3*n} (-1)^(n+k+1) * k * binomial(3*n+k, 2*k). (End)

Extensions

Links edited by Daniel Forgues, Oct 07 2009.

A049614 n! divided by its squarefree kernel.

Original entry on oeis.org

1, 1, 1, 1, 4, 4, 24, 24, 192, 1728, 17280, 17280, 207360, 207360, 2903040, 43545600, 696729600, 696729600, 12541132800, 12541132800, 250822656000, 5267275776000, 115880067072000, 115880067072000, 2781121609728000, 69528040243200000, 1807729046323200000
Offset: 0

Views

Author

Labos Elemer

Keywords

Comments

Also product of composite numbers less than or equal to n. - Benoit Cloitre, Aug 18 2002
Also n! divided by n primorial (or n!/n#). - Cino Hilliard, Mar 26 2006
From Alexander R. Povolotsky and Peter J. C. Moses, Aug 27 2007: (Start)
It appears that a(n) = smallest positive number m such that the sequence b(n) = { m (i^1 + 1!) (i^2 + 2!) ... (i^n + n!) / n! : i >= 0 } takes integral values. [It would be nice to have a proof of this! - N. J. A. Sloane] Cf. A064808 (for n=2), A131682 (for n=3), A131683 (for n=4), A131527 (for n=5), A131684 (for n=6), A131528. See also A129995, A131685. (End)
It appears that every term > 4 is divisible by 24. - Alexander R. Povolotsky, Oct 18 2007
The above comment is correct since each term divides the next. - Charles R Greathouse IV, Jan 16 2012
When n is not a prime number, then a(n)=m*n, where m is some integer >0; such a(n) make up the A036691 Otherwise, when n is a prime number, then a(n)=a(k), where k is the largest nonprime number preceding n (kAlexander R. Povolotsky, Aug 21 2012

Examples

			n = 11: 11! = 39916800 = 2310*17280 and 2310=2*3*5*7*11.

Links

Crossrefs

Cf. A000142, A002110, A003418, A034386, A036691, A045948, A048148.

Programs

  • Magma
    A049614:= func< n | n le 1 select 1 else Factorial(n)/(&*[NthPrime(j): j in [1..#PrimesUpTo(n)]]) >;
[A049614(n): n in [0..40]]; // G. C. Greubel, Jul 21 2023
  • Maple
    primorial := n -> mul(k, k=select(isprime, [$1..n]));
A049614 := n -> factorial(n)/primorial(n);
seq(A049614(i),i=0..24); # Peter Luschny, Feb 16 2013
  • Mathematica
    Table[n!/Product[ Prime[i], {i, PrimePi[n]}], {n, 24}]
  • PARI
    a(n)=prod(i=1,n,i^if(isprime(i),0,1))
  • PARI
    a(n)=n!/prod(i=1,primepi(n),prime(i)) \\ Charles R Greathouse IV, Aug 30 2012
  • SageMath
    def A049614(n): return factorial(n)/product(nth_prime(j) for j in range(1,1+prime_pi(n)))
[A049614(n) for n in range(41)] # G. C. Greubel, Jul 21 2023

Formula

a(n) = A000142(n)/A034386(n).

Extensions

Edited by N. J. A. Sloane, Oct 07 2007
Offset set to 0, a(0)=1 prepended to data, Peter Luschny, Feb 16 2013

A064808 a(n) is the (n+1)st (n+2)-gonal number.

Original entry on oeis.org

1, 3, 9, 22, 45, 81, 133, 204, 297, 415, 561, 738, 949, 1197, 1485, 1816, 2193, 2619, 3097, 3630, 4221, 4873, 5589, 6372, 7225, 8151, 9153, 10234, 11397, 12645, 13981, 15408, 16929, 18547, 20265, 22086, 24013, 26049, 28197, 30460, 32841, 35343, 37969, 40722
Offset: 0

Views

Author

Floor van Lamoen, Oct 22 2001

Keywords

Comments

Sum of n terms of the arithmetic progression with first term 1 and common difference n-1. - Amarnath Murthy, Aug 04 2005
a(n) is the sum of (n+1)-th row terms of triangle A144693. - Gary W. Adamson, Sep 19 2008
See also A131685(k) = smallest positive number m such that c(i) = m*(i^1 + 1)*(i^2 + 2)* ... *(i^k+ k) / k! takes integral values for all i>=0: For k=2, A131685(k)=1, which implies that this is a well-defined integer sequence. - Alexander R. Povolotsky, Apr 24 2015

Links

Crossrefs

Main diagonal of A057145.
Row sums of A076110.
Cf. A144693. - Gary W. Adamson, Sep 19 2008
Cf. A000096, A002378, A006002, A006003.

Programs

Formula

a(n) = (n+1)*(n^2 + 2)/2.
From Paul Barry, Nov 18 2005: (Start)
a(n) = Sum_{k=0..n} Sum_{j=0..n} (k-(k-1)*C(0, j-k)).
a(n) = A006002(n) - A000096(n-2). (End)
G.f.: (1 - x + 3x^2)/(1 - x)^4. - R. J. Mathar, Jul 07 2009
a(n) = A006003(n+1) - A002378(n). - Rick L. Shepherd, Feb 21 2015
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Feb 21 2015
a(n) = A057145(n+2,n+1). - R. J. Mathar, Jul 28 2016

A129995 a(n) = (n + 1)*(n^2 + 2)*(n^3 + 3)*(n^4 + 4)/4!.

Original entry on oeis.org

1, 5, 165, 4675, 65325, 543456, 3155425, 14146210, 52259625, 166192975, 469090061, 1201490445, 2839166005, 6268589250, 13060542825, 25881747316, 49095506065, 89615392425, 158091087925, 270522770375, 450420100221, 731644012660, 1162094343345, 1808433948150
Offset: 0

Views

Author

Alexander R. Povolotsky, Aug 19 2007, Aug 25 2007

Keywords

Comments

Following my conjecture, computations by Peter J. C. Moses, mediation by Clark Kimberling and helpful comments from George E. Andrews, it is now known that a(n) = (n^1 + 1)*(n^2 + 2)*(n^3 + 3)*...*(n^k + k)/k! is an integer-valued sequence if and only if k belongs to {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21}; this is the case for k=4; see generalization in A131685.

Links

Crossrefs

Cf. A000027 (k=1), A064808 (k=2), A131509 (k=3), this sequence (k=4), A131675 (k=5), ..., A131680 (k=10).
See A131685 for a generalization.

Programs

  • Magma
    [(n^1 + 1)*(n^2 + 2)*(n^3 + 3)*(n^4 + 4)/24: n in [0..30]]; // Vincenzo Librandi, Apr 25 2015
  • Maple
    p:=proc(n,i) mul( n^j+j, j=1..i)/i!; end; [seq(p(n,4),n=0..30)];
seq((n+1)*(n^2+2)*(n^3+3)*(n^4+4)/factorial(4), n = 0 .. 20) # Emeric Deutsch, Aug 26 2007
  • Mathematica
    Table[x = 4; Product[(n^k) + k, {k, x}]/x!, {n, 0, 23}] (* Michael De Vlieger, Apr 24 2015 *)
LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{1,5,165,4675,65325,543456,3155425,14146210,52259625,166192975,469090061},30] (* Harvey P. Dale, Dec 07 2021 *)
  • PARI
    vector(20,n,n--;(n+1)*(n^2+2)*(n^3+3)*(n^4+4)/4!) \\ Derek Orr, Apr 25 2015
  • PARI
    A129995(n)=(n+1)*(n^2+2)*(n^3+3)*(n^4+4)/12 \\ M. F. Hasler, May 02 2015

Formula

G.f.: (1-6x+165x^2+2970x^3+22480x^4+55969x^5+51511x^6+16490x^7+1595x^8+25x^9)/(1-x)^11. - Emeric Deutsch, Aug 26 2007
G.f.: -(1 + x*(-6 + x*(165 + x*(2970 + x*(22480 + x*(55969 + x*(51511 + 5*x*(3298 + x*(319 + 5*x))))))))) / (x - 1)^11. - Peter J. C. Moses, Aug 29 2007

A131509 a(n) = (n + 1)*(n^2 + 2)*(n^3 + 3)/6.

Original entry on oeis.org

1, 4, 33, 220, 1005, 3456, 9709, 23528, 50985, 101260, 187561, 328164, 547573, 877800, 1359765, 2044816, 2996369, 4291668, 6023665, 8303020, 11260221, 15047824, 19842813, 25849080, 33300025, 42461276, 53633529, 67155508, 83407045, 102812280, 125842981
Offset: 0

Views

Author

Alexander R. Povolotsky, Aug 13 2007, Aug 25 2007

Keywords

Comments

See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0. For k=3, A131685(k)=1, which implies that this is a well defined integer sequence. - Alexander R. Povolotsky, May 18 2015

Links

Crossrefs

Cf. A000027 (k=1), A064808 (k=2), this sequence (k=3), A129995 (k=4), A131675 (k=5), ..., A131680 (k=10).

Programs

  • Magma
    [(n^1 + 1)*(n^2 + 2)*(n^3 + 3)/6: n in [0..30]]; // Vincenzo Librandi, Apr 25 2015
  • Maple
    p:=proc(n,i) mul( n^j+j, j=1..i)/i!; end; [seq(p(n,3),n=0..30)];
seq((1/6)*(n+1)*(n^2+2)*(n^3+3),n=0..25); # Emeric Deutsch, Aug 23 2007
  • Mathematica
    Table[x = 3; Product[(n^k) + k, {k, x}]/6, {n, 0, 27}] (* Michael De Vlieger, Apr 24 2015 *)
LinearRecurrence[{7,-21,35,-35,21,-7,1},{1,4,33,220,1005,3456,9709},40] (* Harvey P. Dale, Oct 18 2016 *)
  • Maxima
    A131509(n):=(n^1 + 1)*(n^2 + 2)*(n^3 + 3)/6$
makelist(A131509(n),n,0,30); /* Martin Ettl, Nov 03 2012 */
  • PARI
    vector(20,n,n--;(n+1)*(n^2+2)*(n^3+3)/3!) \\ Derek Orr, Apr 25 2015
  • PARI
    A131509(n)=(n+1)*(n^2+2)*(n^3+3)/6 \\ M. F. Hasler, May 02 2015

Formula

G.f.: (1 -3x +26x^2 +38x^3 +53x^4 +5x^5)/(1-x)^7. - Emeric Deutsch, Aug 23 2007

Extensions

Corrected and extended by R. J. Mathar and Emeric Deutsch, Aug 21 2007

A131675 a(n) = (Product_{i=1..5} n^i+i)/5!.

Original entry on oeis.org

1, 6, 1221, 231880, 13443885, 340203456, 4910472385, 47565216504, 342540938025, 1962871989130, 9382270310061, 38701449021984, 141297910237237, 465502930269300, 1404867737405385, 3930816255364816, 10296122969028753, 25448298063869070, 59744930256741205
Offset: 0

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

See A131685 about well-definedness. - M. F. Hasler, May 02 2015

Links

Programs

  • Magma
    [((n+1)*(n^2+2)*(n^3+3)*(n^4+4)*(n^5+5))/Factorial(5): n in [0..20]]; // Vincenzo Librandi, Apr 25 2015
  • Mathematica
    Table[x = 5; Product[(n^k) + k, {k, x}]/x!, {n, 0, 17}] (* Michael De Vlieger, Apr 24 2015 *)
LinearRecurrence[{16,-120,560,-1820,4368,-8008,11440,-12870,11440,-8008,4368,-1820,560,-120,16,-1},{1,6,1221,231880,13443885,340203456,4910472385,47565216504,342540938025,1962871989130,9382270310061,38701449021984,141297910237237,465502930269300,1404867737405385,3930816255364816},20] (* Harvey P. Dale, Oct 04 2024 *)
  • PARI
    Vec((135*x^14 +86852*x^13 +5864611*x^12 +109724496*x^11 +782427151*x^10 +2468818430*x^9 +3704965659*x^8 +2710222344*x^7 +952834509*x^6 +152249688*x^5 +9878785*x^4 +212504*x^3 +1245*x^2 -10*x +1) / (x -1)^16 + O(x^100)) \\ Colin Barker, Apr 24 2015
  • PARI
    A131675(n,k=5)=prod(i=1,k,(n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131676 (k=6), ..., A131680 (k=10). - M. F. Hasler, May 02 2015

Formula

G.f.: (135*x^14 +86852*x^13 +5864611*x^12 +109724496*x^11 +782427151*x^10 +2468818430*x^9 +3704965659*x^8 +2710222344*x^7 +952834509*x^6 +152249688*x^5 +9878785*x^4 +212504*x^3 +1245*x^2 -10*x +1) / (x -1)^16. - Colin Barker, Apr 24 2015

Extensions

Definition made explicit by M. F. Hasler, May 02 2015

A131680 a(n) = (Product_{i=1..10} n^i+i)/10!.

Original entry on oeis.org

1, 11, 54266008005, 94467113468457039310, 538562285352301951109430061, 102370328298891480707678565453456, 2171004564341130364494477279762016705, 10015112821822553484101305268477882115400, 15057116321451208557735379863635553426467625, 9594364176429126945241161642390324911313805168
Offset: 0

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0: For k=10, A131685(k)=1, which implies that this is a well defined integer sequence. - Alexander R. Povolotsky, Apr 24 2015; corrected by M. F. Hasler, May 02 2015

Links

Programs

  • Magma
    [((n+1)*(n^2+2)*(n^3+3)*(n^4+4)*(n^5+5)*(n^6+6)*(n^7+7)*(n^8+8)*(n^9+9)*(n^10+10))/Factorial(10): n in [0..10]]; // Vincenzo Librandi, Apr 25 2015
  • Mathematica
    Table[x = 10; Product[(n^k) + k, {k, x}]/x!, {n, 0, 9}] (* Michael De Vlieger, Apr 24 2015 *)
  • PARI
    A131680(n,k=10)=prod(i=1,k,(n^i+i))/k! \\ Changing the optional 2nd argument allows one to produce A000027 (k=1), A064808 (k=2), A131509 (k=3), A129995 (k=4), A131675(k=5), ..., A131679 (k=9). - M. F. Hasler, May 02 2015

Extensions

Definition made explicit by M. F. Hasler, May 02 2015

A131676 a(n) = (Product_{i=1..6} n^i+i) / 6!.

Original entry on oeis.org

1, 7, 14245, 28405300, 9191136045, 886286703456, 38188743738145, 932714257963020, 14966184483875625, 173860405001195185, 1563721100613810061, 11427034989921521488, 70319024498214551605, 374482754394635213250, 1763001772206469563945, 7462412915610398239816
Offset: 0

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Links

Crossrefs

Cf. A131685.

Programs

Formula

G.f.: (1 - 15*x + 14322*x^2 + 28091987*x^3 + 8569506575*x^4 + 690621422337*x^5 + 20769948618958*x^6 + 283347184706283*x^7 + 1969675285865562*x^8 + 7493939424807955*x^9 + 16292973927985678*x^10 + 20712738704664489*x^11 + 15498276638623618*x^12 + 6765765599122915*x^13 + 1679542499740050*x^14 + 226176197184209*x^15 + 15278037714093*x^16 + 454493699352*x^17 + 4732512736*x^18 + 10869320*x^19 + 1575*x^20)/(1 - x)^22. - M. F. Hasler, May 02 2015

Extensions

Definition made explicit by M. F. Hasler, May 02 2015

A131679 a(n) = (Product_{i=1..9} n^i+i) / 9!.

Original entry on oeis.org

1, 10, 524816325, 15995379784360900, 5136081211768056707885, 104827108835105429096703456, 359044402823940369662885183425, 354548318931625565271233374406000, 140230322081790179721500725877795625, 27516367648544953143193233240569070880, 3102623679344954347223585172112606310061
Offset: 0

Views

Author

Alexander R. Povolotsky, Sep 15 2007

Keywords

Comments

See A131685 about well-definedness. - M. F. Hasler, May 02 2015

Programs

Extensions

Definition made explicit by M. F. Hasler, May 02 2015
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