A285263 Expansion of Product_{k>=1} ((1-x^(5*k))/(1-x^k))^k.

1, 1, 3, 6, 13, 23, 47, 83, 154, 269, 474, 809, 1387, 2313, 3859, 6330, 10341, 16680, 26790, 42586, 67375, 105731, 165097, 256052, 395248, 606501, 926502, 1408048, 2130788, 3209643, 4815595, 7194875, 10709843, 15881236, 23467805, 34556842, 50720003, 74200845

Offset: 0

Author

Vaclav Kotesovec, Apr 15 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(3 * 2^(-2/3) * ((1-1/m^2)*Zeta(3))^(1/3) * n^(2/3)) * ((1-1/m^2)*Zeta(3))^(1/6) / (2^(1/3) * sqrt(3*Pi) * m^(1/12) * n^(2/3)).

Links

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

Crossrefs

Cf. A026007 (m=2), A263346 (m=3), A285262 (m=4).

Cf. A285246.

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(2^(1/3) * 3^(4/3) * 5^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * (2*Zeta(3))^(1/6) / (3^(1/3) * 5^(5/12) * sqrt(Pi) * n^(2/3)).