A320968 - OEIS

%I #24 Oct 30 2018 03:36:45

%S 1,2,4,10,18,34,64,110,188,320,524,846,1358,2130,3308,5102,7750,11674,

%T 17468,25862,38022,55558,80532,116034,166284,236784,335416,472868,

%U 663146,925762,1286920,1780962,2454792,3370806,4610656,6284090,8535868,11554834,15591564

%N 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 Vaclav Kotesovec, <a href="/A320968/b320968.txt">Table of n, a(n) for n = 0..10000</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 * A320098(n).

%F Expansion of Product_{k>0} eta(q^(2*k))^3 / (eta(q^k)^2*eta(q^(4*k))).

%F Expansion of Product_{k>0} 1/theta_4(q^(2*k-1)).

%t 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 (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 Cf. A000122, A002448, A080054 ((theta_3(q^k)/theta_4(q^k))^(1/2)), A320098, A320967, A320992.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 25 2018