A295179 Expansion of Product_{k>=1} 1/(1 - x^k)^(3*k*(k-1)/2+1).

1, 1, 5, 15, 44, 115, 312, 790, 2004, 4908, 11885, 28170, 65987, 152079, 346560, 779808, 1736460, 3825995, 8351733, 18064545, 38747740, 82443251, 174096564, 364991008, 759989218, 1572126699, 3231929735, 6604498620, 13419469596, 27117216441, 54508611399, 109013531864, 216956853105

Offset

0,3

Comments

Euler transform of the centered triangular numbers (A005448).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..4000

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/(1 - x^k)^A005448(k).

a(n) ~ exp(-3*Zeta'(-1)/2 + 7*Zeta(3) / (8*Pi^2) - 225*Zeta(3)^3 / (2*Pi^8) + (Pi / (3*2^(3/4)) - 45*Zeta(3)^2 / (2^(7/4) * Pi^5)) * (5*n)^(1/4) - (3*sqrt(5/2) * Zeta(3) / Pi^2) * sqrt(n) + (2^(7/4)*Pi / (3*5^(1/4))) * n^(3/4)) / (2^(71/32) * 5^(7/32) * Pi^(1/8) * n^(23/32)). - Vaclav Kotesovec, Nov 16 2017

Mathematica

nmax = 32; CoefficientList[Series[Product[1/(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[d (3 d (d - 1)/2 + 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 32}]

Crossrefs

Cf. A000294, A000335, A005448, A295180.

Sequence in context: A025471 A064453 A059251 * A274020 A109952 A146632

Adjacent sequences: A295176 A295177 A295178 * A295180 A295181 A295182

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 16 2017

Status

approved