A282920 Expansion of Product_{k>=1} (1 - x^(7*k))^8/(1 - x^k)^9 in powers of x.

1, 9, 54, 255, 1035, 3753, 12483, 38701, 113193, 315013, 839802, 2155905, 5352252, 12894426, 30233558, 69160869, 154677325, 338822547, 728084435, 1536931932, 3190959918, 6523084815, 13142291319, 26118847655, 51244059231, 99322878506, 190306301025

Offset: 0

Author

Seiichi Manyama, Feb 24 2017

Keywords

• nonn

Links

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

Crossrefs

Cf. A282919.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&*[(1 - x^(7*j))^8/(1 - x^j)^9: j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018

Mathematica

nmax = 30; CoefficientList[Series[Product[(1 - x^(7*k))^8 /(1 - x^k)^9, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

PARI

my(N=30,x='x+O('x^N)); Vec(prod(j=1, N, (1 - x^(7*j))^8/(1 - x^j)^9)) \\ G. C. Greubel, Nov 18 2018

Sage

R = PowerSeriesRing(ZZ, 'x')
prec = 30
x = R.gen().O(prec)
s = prod((1 - x^(7*j))^8/(1 - x^j)^9 for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^8/(1 - x^n)^9.

a(n) ~ exp(Pi*sqrt(110*n/21)) * sqrt(55) / (4*sqrt(3) * 7^(9/2) * n). - Vaclav Kotesovec, Nov 10 2017