A278558 Expansion of Product_{n>=1} (1 - x^(5*n))^30/(1 - x^n)^31 in powers of x.

1, 31, 527, 6448, 63240, 526443, 3852742, 25380847, 153068700, 855816380, 4479330091, 22117432019, 103672066076, 463698703204, 1987628351600, 8195086588810, 32603090921532, 125497791966435, 468512597653134, 1699911932127300, 6005651320362628, 20693956328627358

Offset: 0

Author

Seiichi Manyama, Nov 23 2016

Keywords

• nonn

Comments

In general, if m>0 and g.f. = Product_{k>=1} (1 - x^(5*k))^m/(1 - x^k)^(m+1) then a(n) ~ sqrt(4*m+5) * exp(Pi*sqrt(2*(4*m+5)*n/15)) / (4*sqrt(3)*5^((m+1)/2)*n). - Vaclav Kotesovec, Nov 28 2016

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A160460, A278555, A278556, A278557, A278559.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^30/(1 - x^k)^31, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 28 2016 *)

Formula

G.f.: Product_{n>=1} (1 - x^(5*n))^30/(1 - x^n)^31.

A278559(n) = 5^2*63*A160460(n) + 5^5*52*A278555(n-1) + 5^7*63*A278556(n-2) + 5^10*6*A278557(n-3) + 5^12*a(n-4) for n >= 4.

a(n) ~ exp(Pi*5*sqrt(2*n/3)) / (24414062500*sqrt(3)*n). - Vaclav Kotesovec, Nov 28 2016