A285291 Expansion of Product_{k>=1} ((1 + x^k) / (1 + x^(5*k)))^k.

1, 1, 2, 5, 8, 15, 27, 47, 78, 134, 218, 356, 576, 916, 1449, 2268, 3525, 5431, 8324, 12652, 19129, 28754, 42974, 63898, 94553, 139241, 204144, 298045, 433328, 627592, 905560, 1301934, 1865362, 2663816, 3791813, 5380911, 7613286, 10740839, 15111141, 21202615

Offset: 0

Author

Vaclav Kotesovec, Apr 16 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(2^(-4/3) * 3^(4/3) * (1-1/m^2)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * ((1-1/m^2)*Zeta(3))^(1/6) / (2^(2/3) * 3^(1/3) * sqrt(Pi) * n^(2/3)).

Links

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

Crossrefs

Cf. A285289 (m=2), A263345 (m=3), A285290 (m=4).

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1+x^(5*k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2^(-1/3) * 3^(5/3) * 5^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (5^(1/3) * 6^(1/6) * sqrt(Pi) * n^(2/3)).