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Expansion of (Sum x^(n^2), n = -inf .. inf )^(-16).
1

%I #18 Sep 20 2018 09:05:56

%S 1,-32,544,-6528,61984,-495040,3453312,-21581568,123040288,-648624288,

%T 3194776000,-14823993472,65231647104,-273714726080,1100198199040,

%U -4252621927680,15859616674336,-57229459033664

%N Expansion of (Sum x^(n^2), n = -inf .. inf )^(-16).

%H Seiichi Manyama, <a href="/A004417/b004417.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 = 16 for this sequence. - _Vaclav Kotesovec_, Aug 18 2015

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

%F G.f.: 1/theta_3(x)^16, 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)^16. (End)

%t nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^16, {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)^16) \\ _Altug Alkan_, Sep 20 2018

%Y Cf. A000122, A000152.

%K sign

%O 0,2

%A _N. J. A. Sloane_