A285932 -id:A285932 - cp's OEIS frontend

A285928 Expansion of (Product_{k>0} (1 - x^(5*k)) / (1 - x^k))^5 in powers of x.

1, 5, 20, 65, 190, 501, 1240, 2890, 6440, 13775, 28502, 57205, 111880, 213670, 399620, 733128, 1321850, 2345340, 4100700, 7072520, 12045005, 20272465, 33746060, 55595635, 90706390, 146638756, 235016940, 373580735, 589238640, 922537655, 1434232510, 2214817165

Offset: 0

Seiichi Manyama, Apr 28 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} ((1 - x^(m*k)) / (1 - x^k))^m, then a(n, m) ~ exp(Pi*sqrt(2*(m-1)*n/3)) * (m-1)^(1/4) / (2^(5/4) * 3^(1/4) * m^(m/2) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

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

(Product_{k>0} (1 - x^(m*k)) / (1 - x^k))^m: A022567 (m=2), A285927 (m=3), A093160 (m=4), this sequence (m=5).

Cf. A035959, A285932.

nmax = 40; CoefficientList[Series[Product[((1 - x^(5*k)) / (1 - x^k))^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 30 2017 *)

a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A116073(k)*a(n-k) for n > 0.

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(1/4) * 5^(5/2) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

cp's OEIS Frontend

A285928 Expansion of (Product_{k>0} (1 - x^(5*k)) / (1 - x^k))^5 in powers of x.

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