A004404 Expansion of 1 / (Sum_{n=-oo..oo} x^(n^2))^3.

1, -6, 24, -80, 234, -624, 1552, -3648, 8184, -17654, 36816, -74544, 147056, -283440, 535008, -990912, 1803882, -3232224, 5707624, -9943536, 17106960, -29088352, 48922320, -81438528, 134261584, -219336630, 355242288

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..4000 from Robert Israel)
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.

Crossrefs

Cf. A001934, A004405-A004425, A015128.

Programs

Maple

S:= series(1/JacobiTheta3(0,x)^3,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Dec 29 2015

Mathematica

nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

Formula

a(n) ~ (-1)^n * 3*exp(Pi*sqrt(3*n)) / (64*n^(3/2)) * (1 - sqrt(3)/(Pi*sqrt(n))). - Vaclav Kotesovec, Aug 18 2015, extended Jan 16 2017