A293306 Expansion of (eta(q)*eta(q^3))/eta(q^2)^2 in powers of q.

1, -1, 1, -3, 4, -5, 6, -9, 13, -16, 20, -27, 36, -44, 54, -69, 88, -107, 130, -162, 200, -240, 288, -351, 426, -507, 602, -723, 864, -1019, 1200, -1422, 1681, -1968, 2300, -2700, 3160, -3674, 4266, -4965, 5768, -6665, 7692, -8892, 10260, -11792, 13536, -15552

Offset: 0

Author

Seiichi Manyama, Oct 05 2017

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See p. 15.

Crossrefs

Main diagonal of A293305.

Cf. A122792.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^k) * (1 - x^(3*k)) / (1 - x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2017 *)

Formula

G.f.: Product_{i>0} (1 + Sum_{j>0} (-1)^j*j*q^(j*i)).

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)). - Vaclav Kotesovec, Oct 05 2017