A022574 Expansion of Product_{m>=1} (1+x^m)^9.

1, 9, 45, 174, 576, 1701, 4614, 11709, 28125, 64525, 142353, 303552, 628251, 1266273, 2492352, 4801578, 9071973, 16837893, 30744649, 55296000, 98070633, 171683463, 296919081, 507695670, 858866880, 1438391232, 2386178649, 3923081006, 6395198049, 10341173376, 16593811467

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

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

Crossrefs

Cf. A000009.

Column k=9 of A286335.

Programs

Magma

Coefficients(&*[(1+x^m)^9:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 26 2018

Mathematica

nmax=50; CoefficientList[Series[Product[(1+q^m)^9,{m,1,nmax}],{q,0,nmax}],q] (* Vaclav Kotesovec, Mar 05 2015 *)

PARI

m=50; q='q+O('q^m); Vec(prod(n=1,m,(1+q^n)^9)) \\ G. C. Greubel, Feb 26 2018

Formula

a(n) ~ 3^(1/4) * exp(Pi * sqrt(3*n)) / (64 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

a(0) = 1, a(n) = (9/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017

G.f.: exp(9*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018