A320098 Expansion of Product_{k>0} 1/theta_3(q^(2*k-1)), where theta_3() is the Jacobi theta function.

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

Offset: 0

Author

Seiichi Manyama, Oct 05 2018

Keywords

• sign

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

Cf. A000122, A320068, A320078.

Programs

Mathematica

nmax = 50; CoefficientList[Series[1/Product[EllipticTheta[3, 0, x^(2*k-1)], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*j*(2*k - 1)))^2/((1 - x^((2*k - 1)*j))*(1 + x^((2*k - 1)*j))^3), {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2018 *)

PARI

q='q+O('q^80); Vec(1/prod(k=1,50, eta(q^(2*(2*k-1)))^5/( eta(q^(2*k-1))* eta(q^(4*(2*k-1))))^2 ) ) \\ G. C. Greubel, Oct 29 2018

Formula

Convolution inverse of A320078.

Expansion of Product_{k>0} (eta(q^(2*k-1))*eta(q^(4*(2*k-1))))^2 / eta(q^(2*(2*k-1)))^5.