A295180 Expansion of Product_{k>=1} (1 + x^k)^(3*k*(k-1)/2+1).

1, 1, 4, 14, 35, 96, 242, 609, 1483, 3565, 8376, 19389, 44254, 99584, 221470, 486810, 1058914, 2280519, 4866492, 10294313, 21598679, 44966391, 92930485, 190721585, 388828094, 787710401, 1586166758, 3175548134, 6322372729, 12520759979, 24669499432, 48367447687, 94381633962, 183331308393

Offset

0,3

Comments

Weigh transform of the centered triangular numbers (A005448).

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -(3*n*(n-1)/2+1), g(n) = -1. - Seiichi Manyama, Nov 16 2017

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Centered Triangular Number

Index entries for sequences related to centered polygonal numbers

Formula

G.f.: Product_{k>=1} (1 + x^k)^A005448(k).

a(n) ~ exp(15*Zeta(3) / (28*Pi^2) - 6075*Zeta(3)^3 / (98*Pi^8) + (Pi/6 - 405*Zeta(3)^2 / (28*Pi^5)) * (5*n/7)^(1/4) - (9*sqrt(5/7) * Zeta(3) / (2*Pi^2)) * sqrt(n) + (2*Pi * (7/5)^(1/4)/3) * n^(3/4)) * 7^(1/8) / (2^(19/8) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Nov 16 2017

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d*(3*d*(d-1)/2+1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 16 2017

Mathematica

nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(3 k (k - 1)/2 + 1), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (3 d (d - 1)/2 + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Crossrefs

Cf. A005448, A028377, A258343, A295179.

Sequence in context: A011852 A307260 A288678 * A305906 A177110 A213045

Adjacent sequences: A295177 A295178 A295179 * A295181 A295182 A295183

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 16 2017

Status

approved