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%I #20 Oct 26 2018 09:27:00
%S 1,-2,0,-2,6,-2,4,-6,8,-16,8,-14,26,-26,24,-30,58,-50,60,-78,90,-118,
%T 104,-138,192,-224,204,-268,366,-354,412,-474,596,-694,724,-818,1052,
%U -1162,1176,-1470,1756,-1918,2052,-2434,2814,-3168,3396,-3806,4674,-5124,5396
%N Expansion of (Product_{k>0} theta_4(q^k)/theta_3(q^k))^(1/2), where theta_3() and theta_4() are the Jacobi theta functions.
%H Vaclav Kotesovec, <a href="/A320992/b320992.txt">Table of n, a(n) for n = 0..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F a(n) = (-1)^n * A320078(n).
%F Expansion of Product_{k>0} (eta(q^k)^2*eta(q^(4*k))) / eta(q^(2*k))^3.
%F Expansion of Product_{k>0} theta_4(q^(2*k-1)).
%F a(n) ~ (-1)^n * (log(2))^(1/4) * exp(Pi*sqrt(n*log(2)/2)) / (4*n^(3/4)). - _Vaclav Kotesovec_, Oct 26 2018
%t nmax = 60; CoefficientList[Series[Product[Sqrt[EllipticTheta[4, 0, x^k] / EllipticTheta[3, 0, x^k]], {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 26 2018 *)
%Y Convolution inverse of A320968.
%Y Cf. A000122, A002448, A320078, A320970.
%K sign
%O 0,2
%A _Seiichi Manyama_, Oct 26 2018
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