A022584 Expansion of Product_{m>=1} (1+x^m)^19.

1, 19, 190, 1349, 7676, 37278, 160417, 626924, 2263698, 7647652, 24405633, 74120672, 215505334, 602763220, 1628328880, 4262845643, 10845598563, 26882001287, 65048680364, 153950675585, 356936640088, 811869015895, 1813912504439, 3985419541978, 8619872682020, 18369414409148

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

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

Crossrefs

Column k=19 of A286335.

Programs

Magma

Coefficients(&*[(1+x^m)^19:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Feb 25 2018

Mathematica

nmax=50; CoefficientList[Series[Product[(1+q^m)^19,{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)^19)) \\ G. C. Greubel, Feb 25 2018

Formula

a(n) ~ (19/3)^(1/4) * exp(Pi * sqrt(19*n/3)) / (2048 * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

a(0) = 1, a(n) = (19/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 04 2017