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A225915 Expansion of (k(q) / 4)^4 in powers of q where k() is a Jacobi elliptic function.

%I A225915 #17 Aug 10 2018 01:26:33
%S A225915 1,-16,152,-1088,6444,-33184,153152,-646528,2533070,-9311664,32387616,
%T A225915 -107299904,340436664,-1039026144,3061896704,-8739810688,24229115109,
%U A225915 -65390485328,172155210320,-442928464640,1115433685796,-2753362613984,6670224790272,-15876957230848
%N A225915 Expansion of (k(q) / 4)^4 in powers of q where k() is a Jacobi elliptic function.
%H A225915 G. C. Greubel, <a href="/A225915/b225915.txt">Table of n, a(n) for n = 2..2500</a>
%F A225915 Expansion of (eta(q) * eta(q^4)^2 / eta(q^2)^3)^16 in powers of q.
%F A225915 Euler transform of period 4 sequence [-16, 32, -16, 0, ...].
%F A225915 G.f.: q^2 * (Product_{k>0} (1 + q^(2*k)) / (1 + q^(2*k - 1)))^16.
%F A225915 Convolution square of A005798.
%F A225915 a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n)) / (65536 * 2^(3/4) * n^(3/4)). - _Vaclav Kotesovec_, Nov 17 2017
%e A225915 G.f. = q^2 - 16*q^3 + 152*q^4 - 1088*q^5 + 6444*q^6 - 33184*q^7 + 153152*q^8 + ...
%t A225915 a[ n_] := SeriesCoefficient[ (InverseEllipticNomeQ[  q] / 16)^2, {q, 0, n}];
%t A225915 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q] / EllipticTheta[ 2, 0, q^(1/2)])^16, {q, 0, n}];
%t A225915 a[ n_] := SeriesCoefficient[ q ( Product[ 1 - q^k, {k, 4, n - 1, 4}]/
%t A225915 Product[ 1 - (-q)^k, {k, n - 1}])^16, {q, 0, n}];
%o A225915 (PARI) {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3)^16, n))};
%Y A225915 Cf. A001938, A005798, A079006, A083365.
%K A225915 sign
%O A225915 2,2
%A A225915 _Michael Somos_, May 20 2013