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A320968 Expansion of (Product_{k>0} theta_3(q^k)/theta_4(q^k))^(1/2), where theta_3() and theta_4() are the Jacobi theta functions.

%I A320968 #25 Feb 16 2025 08:33:57
%S A320968 1,2,4,10,18,34,64,110,188,320,524,846,1358,2130,3308,5102,7750,11674,
%T A320968 17468,25862,38022,55558,80532,116034,166284,236784,335416,472868,
%U A320968 663146,925762,1286920,1780962,2454792,3370806,4610656,6284090,8535868,11554834,15591564
%N A320968 Expansion of (Product_{k>0} theta_3(q^k)/theta_4(q^k))^(1/2), where theta_3() and theta_4() are the Jacobi theta functions.
%H A320968 Vaclav Kotesovec, <a href="/A320968/b320968.txt">Table of n, a(n) for n = 0..10000</a>
%H A320968 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A320968 a(n) = (-1)^n * A320098(n).
%F A320968 Expansion of Product_{k>0} eta(q^(2*k))^3 / (eta(q^k)^2*eta(q^(4*k))).
%F A320968 Expansion of Product_{k>0} 1/theta_4(q^(2*k-1)).
%t A320968 CoefficientList[Series[1/Product[EllipticTheta[4, 0, q^(2*k - 1)], {k, 1, 50}], {q, 0, 80}], q] (* _G. C. Greubel_, Oct 29 2018 *)
%o A320968 (PARI) q='q+O('q^80); Vec(prod(k=1,50, eta(q^(2*k))^3/(eta(q^k)^2* eta(q^(4*k))) )) \\ _G. C. Greubel_, Oct 29 2018
%Y A320968 Cf. A000122, A002448, A080054 ((theta_3(q^k)/theta_4(q^k))^(1/2)), A320098, A320967, A320992.
%K A320968 nonn
%O A320968 0,2
%A A320968 _Seiichi Manyama_, Oct 25 2018