A320070 Expansion of 1/(theta_3(q) * theta_3(q^2) * theta_3(q^3)), where theta_3() is the Jacobi theta function.

1, -2, 2, -6, 14, -20, 32, -60, 98, -150, 232, -360, 558, -828, 1196, -1776, 2614, -3700, 5238, -7480, 10516, -14592, 20180, -27832, 38216, -51970, 70184, -94842, 127612, -170140, 226164, -300324, 396754, -521520, 683484, -893432, 1164330, -1511188, 1954756, -2524188

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Simon Plouffe, Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 19/24, marked 212.

Formula

Convolution inverse of A029594.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (4*sqrt(6)*n^(3/2)). - Vaclav Kotesovec, Oct 05 2018

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (32/9) * Pi^(15/4) * 3^(1/2) * Gamma(7/8)^8 * (2-sqrt(2))^(1/2) / (1+3^(1/2)) / Gamma(5/8)^8 / Gamma(7/12)^3 / (17/12+sqrt(2)) / Gamma(11/12)^2 / Gamma(2/3) = A389046. - Simon Plouffe, Sep 22 2025

Mathematica

CoefficientList[Series[1/Product[EllipticTheta[3, 0, q^k], {k, 1, 3}], {q, 0, 80}], q] (* G. C. Greubel, Oct 29 2018 *)

Prog

(PARI) q='q+O('q^80); Vec(1/prod(k=1, 3, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )) \\ G. C. Greubel, Oct 29 2018

Crossrefs

Cf. A000122, A029594, A320068, A389046.

Sequence in context: A019100 A019101 A233230 * A319866 A266007 A051890

Adjacent sequences: A320067 A320068 A320069 * A320071 A320072 A320073

Keyword

sign

Author

Seiichi Manyama, Oct 05 2018

Status

approved