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.

1, 2, 4, 10, 18, 34, 64, 110, 188, 320, 524, 846, 1358, 2130, 3308, 5102, 7750, 11674, 17468, 25862, 38022, 55558, 80532, 116034, 166284, 236784, 335416, 472868, 663146, 925762, 1286920, 1780962, 2454792, 3370806, 4610656, 6284090, 8535868, 11554834, 15591564

Offset: 0

Author

Seiichi Manyama, Oct 25 2018

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

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

Programs

Mathematica

CoefficientList[Series[1/Product[EllipticTheta[4, 0, q^(2*k - 1)], {k, 1, 50}], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)

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

Formula

a(n) = (-1)^n * A320098(n).

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

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