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Expansion of ( Sum_{n = -infinity..infinity} x^(n^2) )^(-6).
2

%I #24 Sep 20 2018 09:17:09

%S 1,-12,84,-448,2004,-7896,28224,-93312,289236,-848972,2377704,

%T -6391872,16571968,-41599320,101430144,-240877440,558440916,

%U -1266406680,2814053908,-6136337088,13148606184,-27717527552

%N Expansion of ( Sum_{n = -infinity..infinity} x^(n^2) )^(-6).

%H Seiichi Manyama, <a href="/A004407/b004407.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ (-1)^n * 3^(7/4)*exp(Pi*sqrt(6*n)) / (256*2^(3/4)*n^(9/4)). - _Vaclav Kotesovec_, Aug 18 2015

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

%F G.f.: 1/theta_3(x)^6, 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)^6. (End)

%t nmax = 30; CoefficientList[Series[Product[((1 + (-x)^k)/(1 - (-x)^k))^6, {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)^6) \\ _Altug Alkan_, Sep 20 2018

%Y Cf. A000122, A000141.

%K sign

%O 0,2

%A _N. J. A. Sloane_