A285340 Expansion of Product_{k>=0} (1 + x^(5*k+4))^(5*k+4).

1, 0, 0, 0, 4, 0, 0, 0, 6, 9, 0, 0, 4, 36, 14, 0, 1, 54, 92, 19, 0, 36, 228, 202, 24, 9, 272, 702, 358, 29, 158, 1168, 1696, 598, 70, 1027, 3810, 3605, 904, 501, 4600, 10196, 6898, 1408, 3078, 15805, 24104, 12242, 2838, 14103, 46090, 51376, 20566, 9443, 51682

Offset: 0

Author

Seiichi Manyama, Apr 17 2017

Keywords

• nonn

Comments

In general, if m > 1 and g.f. = Product_{k>=1} (1 + x^(m*k-1))^(m*k-1), then a(n, m) ~ exp(2^(-4/3) * 3^(4/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(1/6 + 1/(2*m) + m/12) * 3^(1/3) * m^(1/6) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Apr 17 2017

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000

Crossrefs

Product_{k>=0} (1 + x^(m*k+m-1))^(m*k+m-1): A262736 (m=2), A262948 (m=3), A285339 (m=4), this sequence (m=5).

Cf. A285214, A285338.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 + x^(5*k-1))^(5*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 17 2017 *)

Formula

a(n) ~ exp(2^(-4/3) * 3^(4/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(41/60) * 3^(1/3) * 5^(1/6) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Apr 17 2017