A285461 Expansion of Product_{k>=1} ((1 + x^(5*k)) / (1 - x^k))^k.

1, 1, 3, 6, 13, 25, 49, 89, 166, 295, 526, 909, 1571, 2657, 4475, 7432, 12257, 20000, 32436, 52126, 83285, 132057, 208221, 326202, 508372, 787777, 1214828, 1863932, 2847020, 4328765, 6554359, 9882795, 14843999, 22210386, 33112817, 49192218, 72834243

Offset: 0

Author

Vaclav Kotesovec, Apr 19 2017

Keywords

• nonn

Comments

In general, if m >= 1 and g.f. = Product_{k>=1} ((1 + x^(m*k)) / (1 - x^k))^k, then a(n, m) ~ exp(1/12 + 3 * 2^(-4/3) * (4 + 3/m^2)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * (4 + 3/m^2)^(7/36) * Zeta(3)^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A156616 (m=1), A285462 (m=2), A285447 (m=3), A285460 (m=4).

Cf. A024789.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[((1+x^(5*k))/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(1/12 + 3 * 2^(-4/3) * 5^(-2/3) * (103*Zeta(3))^(1/3) * n^(2/3)) * (103*Zeta(3))^(7/36) / (A * 2^(7/9) * 5^(7/18) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.