A262923 Expansion of Product_{k>=1} 1 / ((1-x^(3*k-1))^(3*k-1) * (1-x^(3*k-2))^(3*k-2)).

1, 1, 3, 3, 10, 15, 27, 44, 79, 128, 211, 331, 549, 843, 1338, 2061, 3195, 4851, 7384, 11104, 16696, 24774, 36817, 54173, 79560, 116067, 168880, 244293, 352480, 506012, 724531, 1032762, 1468271, 2079525, 2937102, 4134399, 5804795, 8124459, 11342952, 15791650

Offset: 0

Author

Vaclav Kotesovec, Oct 04 2015

Keywords

• nonn

Comments

Convolution of A262946 and A262947.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.

Crossrefs

Cf. A262883, A262876, A262877, A262924, A035386, A035382, A262946, A262947.

Programs

Mathematica

nmax=60; CoefficientList[Series[Product[1/((1 - x^(3*k-1))^(3*k-1) * (1 - x^(3*k-2))^(3*k-2)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(-1/6 + 3^(2/3)*(Zeta(3)/2)^(1/3) * n^(2/3)) * A^2 * Zeta(3)^(1/9) / (2^(5/18) * 3^(31/36) * sqrt(Pi) * n^(11/18)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.