A004414 Expansion of (Sum_{n=-inf..inf} x^(n^2))^(-13).

1, -26, 364, -3640, 29094, -197288, 1177176, -6333184, 31258604, -143374530, 617193304, -2513060264, 9739727816, -36115518376, 128680223152, -442158402816, 1469734751654, -4738671343952, 14853923411652

Offset

0,2

Links

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

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 = 13 for this sequence. - Vaclav Kotesovec, Aug 18 2015

From Ilya Gutkovskiy, Sep 20 2018: (Start)

G.f.: 1/theta_3(x)^13, where theta_3() is the Jacobi theta function.

G.f.: Product_{k>=1} 1/((1 - x^(2*k))*(1 + x^(2*k-1))^2)^13. (End)

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1 / Pi^(13/4)) * Gamma(3/4)^13 = A388128. - Simon Plouffe, Sep 15 2025

Mathematica

nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^13, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)

Crossrefs

Cf. A000122, A276285.

Sequence in context: A010978 A022590 A364010 * A125461 A022654 A183187

Adjacent sequences: A004411 A004412 A004413 * A004415 A004416 A004417

Keyword

sign

Author

N. J. A. Sloane

Status

approved