A285246 Expansion of Product_{k>=1} (1 - x^(5*k))^(5*k) / (1 - x^k)^k.

1, 1, 3, 6, 13, 19, 43, 71, 130, 217, 380, 619, 1049, 1685, 2757, 4404, 7027, 11014, 17326, 26820, 41488, 63514, 96970, 146808, 221659, 332212, 496439, 737535, 1091938, 1608564, 2361929, 3452736, 5031138, 7302373, 10566038, 15234196, 21900182, 31380435

Offset: 0

Author

Seiichi Manyama, Apr 15 2017

Keywords

• nonn

Comments

In general, if m > 1 and g.f. = Product_{k>=1} (1 - x^(m*k))^(m*k)/((1 - x^k)^k), then a(n, m) ~ exp(1/12 - m/12 + 3 * 2^(-2/3) * (1-1/m)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(-(m+11)/36) * A^(m-1) * (m-1)^((7-m)/36) * m^(-(2*m+7)/36) * Zeta(3)^((7-m)/36) * n^((m-25)/36) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 16 2017

Links

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

Crossrefs

Product_{k>=1} (1 - x^(m*k))^(m*k)/(1 - x^k)^k: A262811 (m=2), A262923 (m=3), A285215 (m=4), this sequence (m=5).

Cf. A285263, A285285.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[1 / ((1-x^(5*k+1))^(5*k+1) * (1-x^(5*k+2))^(5*k+2) * (1-x^(5*k+3))^(5*k+3) * (1-x^(5*k+4))^(5*k+4)), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^(5*k)/((1 - x^k)^k), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)

PARI

x='x+O('x^100); Vec(prod(k=0, 100, 1 / ((1 - x^(5*k + 1))^(5*k + 1)*(1 - x^(5*k + 2))^(5*k + 2)*(1 - x^(5*k + 3))^(5*k + 3)*(1 - x^(5*k + 4))^(5*k + 4)))) \\ Indranil Ghosh, Apr 15 2017

Formula

G.f.: Product_{k>=0} 1 / ((1-x^(5*k+1))^(5*k+1) * (1-x^(5*k+2))^(5*k+2) * (1-x^(5*k+3))^(5*k+3) * (1-x^(5*k+4))^(5*k+4)).

a(n) ~ exp(-1/3 + 3*(Zeta(3)/5)^(1/3)*n^(2/3)) * A^4 * Zeta(3)^(1/18) / (2^(1/3) * 5^(17/36) * sqrt(3*Pi) * n^(5/9)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 16 2017