A004416 - OEIS

%I #23 Sep 15 2025 19:45:07

%S 1,-30,480,-5440,48930,-371136,2464320,-14688000,80001120,-403533790,

%T 1904433984,-8477603520,35829727680,-144548556480,559157308800,

%U -2081866609920,7484792950050,-26057409056640,88057506412320

%N Expansion of (Sum_{n=-inf..inf} x^(n^2))^(-15).

%H Seiichi Manyama, <a href="/A004416/b004416.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ (-1)^n * exp(Pi*sqrt(m*n)) * m^((m+1)/4) / (2^(3*(m+1)/2) * n^((m+3)/4)), set m = 15 for this sequence. - _Vaclav Kotesovec_, Aug 18 2015

%F From _Ilya Gutkovskiy_, Sep 20 2018: (Start)

%F G.f.: 1/theta_3(x)^15, where theta_3() is the Jacobi theta function.

%F G.f.: Product_{k>=1} 1/((1 - x^(2*k))*(1 + x^(2*k-1))^2)^15. (End)

%F Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1 / Pi^(15/4)) * Gamma(3/4)^15 = A388151. - _Simon Plouffe_, Sep 15 2025

%t nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^15, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 18 2015 *)

%o (PARI) q='q+O('q^99); Vec(((eta(q)*eta(q^4))^2/eta(q^2)^5)^15) \\ _Altug Alkan_, Sep 20 2018

%Y Cf. A000122, A276287.

%K sign

%O 0,2

%A _N. J. A. Sloane_