A320078 Expansion of Product_{k>0} theta_3(q^(2*k-1)), where theta_3() is the Jacobi theta function.

1, 2, 0, 2, 6, 2, 4, 6, 8, 16, 8, 14, 26, 26, 24, 30, 58, 50, 60, 78, 90, 118, 104, 138, 192, 224, 204, 268, 366, 354, 412, 474, 596, 694, 724, 818, 1052, 1162, 1176, 1470, 1756, 1918, 2052, 2434, 2814, 3168, 3396, 3806, 4674, 5124, 5396, 6250, 7374, 7898, 8732

Offset: 0

Author

Seiichi Manyama, Oct 05 2018

Keywords

• nonn

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, A033716, A320067, A320098.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Product[EllipticTheta[3, 0, x^(2*k-1)], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 60; CoefficientList[Series[Product[(1 - x^((2*k-1)*j))*(1 + x^((2*k-1)*j))^3/(1 + x^(2*j*(2*k-1)))^2, {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(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

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

a(n) ~ (log(2))^(1/4) * exp(Pi*sqrt(n*log(2)/2)) / (4*n^(3/4)). - Vaclav Kotesovec, Oct 07 2018