%I #25 Oct 30 2018 05:43:31

%S 1,-4,4,-4,20,-28,20,-52,84,-104,156,-180,308,-460,468,-684,1028,

%T -1308,1592,-2084,2940,-3668,4564,-5716,7556,-9912,11484,-14616,19252,

%U -23548,28316,-35188,44724,-54532,65996,-79948,99784,-122796,143972,-175372,216524,-259996,308004,-371140

%N Expansion of Product_{k>0} theta_4(q^k)/theta_3(q^k), where theta_3() and theta_4() are the Jacobi theta functions.

%H Seiichi Manyama, <a href="/A320970/b320970.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 Expansion of Product_{k>0} (eta(q^k)^4*eta(q^(4*k))^2) / eta(q^(2*k))^6.

%F a(n) ~ (-1)^n * exp(Pi*sqrt(log(2)*n)) * (log(2))^(1/4) / (4*n^(3/4)). - _Vaclav Kotesovec_, Oct 26 2018

%t With[{nmax=80}, CoefficientList[Series[Product[EllipticTheta[4, 0, q^k]/EllipticTheta[3, 0, q^k], {k, 1, nmax+2}], {q, 0, nmax}], q]] (* _G. C. Greubel_, Oct 29 2018 *)

%o (PARI) m=80; q='q+O('q^m); Vec(1/prod(k=1,m+2, eta(q^(2*k))^6/( eta(q^k)^4* eta(q^(4*k))^2) )) \\ _G. C. Greubel_, Oct 29 2018

%Y Cf. A000122, A002448, A320068, A320908, A320967.

%K sign

%O 0,2

%A _Seiichi Manyama_, Oct 25 2018