A262883 Expansion of Product_{k>=1} 1/((1-x^(3*k-1))*(1-x^(3*k-2)))^k.

1, 1, 2, 2, 5, 7, 10, 15, 24, 33, 49, 68, 100, 136, 193, 267, 370, 501, 690, 928, 1260, 1687, 2265, 3007, 4006, 5289, 6987, 9163, 12033, 15698, 20469, 26572, 34470, 44510, 57442, 73861, 94852, 121439, 155287, 198007, 252165, 320335, 406396, 514410, 650288

Offset: 0

Author

Vaclav Kotesovec, Oct 04 2015

Keywords

• nonn

Comments

Convolution of A262876 and A262877.

Links

• Vaclav Kotesovec and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2001 terms 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. A262876, A262877, A262884, A262923.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(irem(d+3, 3, 'r')=0, 0, r), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 05 2015

Mathematica

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

Formula

a(n) ~ exp(-1/18 - Pi^4/(864*Zeta(3)) + (3*Zeta(3)/2)^(1/3) * n^(2/3) + Pi^2 * n^(1/3) / (2^(5/3)*3^(4/3)*Zeta(3)^(1/3))) * A^(2/3) * Gamma(4/3)^(1/3) * Zeta(3)^(7/54) / (2^(11/27) * 3^(79/108) * Pi^(2/3) * n^(17/27)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.