A282932 Expansion of Product_{k>=1} (1 - x^(7*k))^56/(1 - x^k)^57 in powers of x.

1, 57, 1710, 35815, 586815, 7997157, 94175267, 983458849, 9279004863, 80218101555, 642408637594, 4807304399931, 33855173217278, 225702273908048, 1431470152072364, 8673471170235715, 50389686887219910, 281575909008910196, 1517580284619183809

Offset: 0

Author

Seiichi Manyama, Feb 24 2017

Keywords

• nonn

Comments

In general, if m >= 1 and g.f. = Product_{k>=1} (1 - x^(7*k))^m / (1 - x^k)^(m+1), then a(n) ~ exp(Pi*sqrt((2*(6*m+7)*n)/21)) * sqrt(6*m+7) / (4*sqrt(3) * 7^((m+1)/2) * n). - Vaclav Kotesovec, Nov 10 2017

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))^56/(1 - x^j)^57: j in [1..m+2]]) )); // G. C. Greubel, Nov 18 2018

Mathematica

nmax = 20; CoefficientList[Series[Product[(1 - x^(7*k))^56/(1 - x^k)^57, {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))^56/(1 - x^j)^57)) \\ G. C. Greubel, Nov 18 2018

Sage

R = PowerSeriesRing(ZZ, 'x')
prec = 30
x = R.gen().O(prec)
s = prod((1 - x^(7*j))^56/(1 - x^j)^57 for j in (1..prec))
print(s.coefficients()) # G. C. Greubel, Nov 18 2018

Formula

G.f.: Product_{n>=1} (1 - x^(7*n))^56/(1 - x^n)^57.

a(n) ~ exp(Pi*sqrt(686*n/21)) * sqrt(343) / (4*sqrt(3) * 7^(57/2) * n). - Vaclav Kotesovec, Nov 10 2017