A285459 Expansion of Product_{k>=1} ((1 + x^k) / (1 - x^(5*k)))^k.

1, 1, 2, 5, 8, 17, 29, 51, 88, 150, 254, 416, 682, 1102, 1765, 2810, 4415, 6897, 10704, 16482, 25251, 38410, 58120, 87480, 130999, 195253, 289612, 427757, 629128, 921590, 1344904, 1955246, 2832608, 4089696, 5885169, 8442269, 12073072, 17214535, 24475417

Offset: 0

Author

Vaclav Kotesovec, Apr 19 2017

Keywords

• nonn

Comments

In general, if m >= 1 and g.f. = Product_{k>=1} ((1 + x^k) / (1 - x^(m*k)))^k, then a(n, m) ~ exp(1/12 + 3 * 2^(-4/3) * (3 + 4/m^2)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * (3 + 4/m^2)^(7/36) * m^(1/12) * 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), A000219 (m=2), A285446 (m=3), A285458 (m=4).

Programs

Mathematica

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

Formula

a(n) ~ exp(1/12 + 3 * 2^(-4/3) * 5^(-2/3) * (79*Zeta(3))^(1/3) * n^(2/3)) * (79*Zeta(3))^(7/36) / (A * 2^(7/9) * 5^(11/36) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.