A004425 Expansion of (Sum x^(n^2), n = -inf .. inf )^(-24).

1, -48, 1200, -20800, 280752, -3142560, 30338880, -259459200, 2003790000, -14178640368, 92960115360, -569803615680, 3289122824000, -17987650183200, 93669997008000, -466466351287680, 2229627536828592, -10261752523778400

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

From Vaclav Kotesovec, Aug 18 2015, extended Jan 16 2017: (Start)

In general, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^m and m>=1, then a(n) ~ exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)) * (1 - (m+3)*(m+1)/(8*Pi*sqrt(m*n))).

If g.f. = Product_{k>=1} ((1+(-x)^k)/(1-(-x)^k))^m and m>=1, then a(n) ~ (-1)^n * exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)) * (1 - (m+3)*(m+1)/(8*Pi*sqrt(m*n))).

(End)

Links

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

Programs

Mathematica

nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^24, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

Formula

a(n) ~ (-1)^n * exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)), set m = 24 for this sequence. - Vaclav Kotesovec, Aug 18 2015