A320068 Expansion of Product_{k>0} 1/theta_3(q^k), where theta_3() is the Jacobi theta function.

1, -2, 2, -6, 12, -18, 30, -50, 84, -132, 198, -306, 476, -706, 1026, -1522, 2234, -3202, 4564, -6506, 9224, -12934, 17982, -24982, 34612, -47496, 64798, -88340, 119944, -161814, 217462, -291562, 389642, -518442, 687222, -908934, 1199040, -1575730, 2064466, -2699378, 3520540

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, A004402, A320067, A320069, A320070, A320968, A320992.

Programs

Magma

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 + x^(2*k*j))^2/((1 - x^(k*j))*(1 + x^(k*j))^3): j in [1..(Floor(2*m/k)+1)]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018

Mathematica

nmax = 50; CoefficientList[Series[1/Product[EllipticTheta[3, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k*j))^2/((1 - x^(k*j))*(1 + x^(k*j))^3), {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2018 *)

PARI

m=50; x='x+O('x^m); Vec(prod(k=1,2*m, prod(j=1,floor(2*m/k)+1, (1 + x^(2*k*j))^2/((1 - x^(k*j))*(1 + x^(k*j))^3) ))) \\ G. C. Greubel, Oct 29 2018

Formula

Convolution inverse of A320067.

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

Expansion of Product_{k>0} theta_4(q^(2*k-1))/theta_4(q^(2*k)), where theta_4() is the Jacobi theta function. - Seiichi Manyama, Oct 26 2018