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A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

%I A294592 #16 Feb 16 2025 08:33:51
%S A294592 1,4,32,304,3072,32024,340352,3666016,39878656,437091892,4819567552,
%T A294592 53401892240,594093969408,6631726263608,74242911364864,
%U A294592 833237193123104,9371924860764160,105614054423502408,1192210691317862048,13478559927485340144,152589996020498655232,1729590806617202662528
%N A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.
%H A294592 Robert Israel, <a href="/A294592/b294592.txt">Table of n, a(n) for n = 0..500</a>
%H A294592 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A294592 Wikipedia, <a href="http://en.wikipedia.org/wiki/Theta_function">Theta function</a>
%F A294592 a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^(2*n).
%F A294592 From _Vaclav Kotesovec_, Nov 05 2017: (Start)
%F A294592 a(n) ~ c * d^n / sqrt(n), where
%F A294592 d = 11.61255065799699699891360038489317237925475956178123836149123386457... and
%F A294592 c = 0.34456510029264878768512693687607064416428117641473856418257649837... (End)
%p A294592 S:= series((JacobiTheta3(0,x)/JacobiTheta4(0,x))^n,x,51):
%p A294592 seq(coeff(S,x,n),n=0..50); # _Robert Israel_, Nov 03 2017
%t A294592 Table[SeriesCoefficient[(EllipticTheta[3, 0, x]/EllipticTheta[4, 0, x])^n, {x, 0, n}], {n, 0, 21}]
%t A294592 Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^(2 n), {k, 0, n}], {x, 0, n}], {n, 0, 21}]
%t A294592 Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^(2 n), {x, 0, n}], {n, 0, 21}]
%t A294592 (* Calculation of constant d: *) 1/r /. FindRoot[{s == EllipticTheta[3, 0, r*s]/EllipticTheta[4, 0, r*s], EllipticTheta[4, 0, r*s] + r*s*Derivative[0, 0, 1][EllipticTheta][4, 0, r*s] == r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s]}, {r, 1/10}, {s, 2}, WorkingPrecision -> 120] (* _Vaclav Kotesovec_, Jan 17 2024 *)
%Y A294592 Cf. A007096, A014969, A014970, A014972, A066535, A080054, A103261, A270919, A289522, A291697.
%K A294592 nonn
%O A294592 0,2
%A A294592 _Ilya Gutkovskiy_, Nov 03 2017