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

0,2

Comments

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

Links

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

Formula

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

Mathematica

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 *)

Crossrefs

(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.

Sequence in context: A195861 A001939 A100534 * A160506 A277212 A160528

Adjacent sequences: A285925 A285926 A285927 * A285929 A285930 A285931

Keyword

nonn

Author

Seiichi Manyama, Apr 28 2017

Status

approved