A022596 Expansion of Product_{m>=1} (1+q^m)^32.

1, 32, 528, 6016, 53384, 393920, 2517824, 14329600, 74059812, 352722720, 1565583648, 6533812352, 25823152256, 97218393280, 350348856704, 1213526698240, 4054279504266, 13103911398400, 41081428394096, 125210147216000, 371754750363712, 1077136199182976, 3050503922469440

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

In general, for k > 0, if g.f. = Product_{m>=1} (1+q^m)^k, then a(n) ~ k^(1/4) * exp(Pi * sqrt(k*n/3)) / (2^((k+3)/2) * 3^(1/4) * n^(3/4)) * (1 + (Pi*k^(3/2) / (48*sqrt(3)) - 3^(3/2) / (8*Pi*sqrt(k))) / sqrt(n)). - Vaclav Kotesovec, Mar 05 2015, extended Jan 16 2017

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000

Crossrefs

Column k=32 of A286335.

Programs

Magma

Coefficients(&*[(1+x^m)^32:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // G. C. Greubel, Mar 20 2018

Mathematica

nmax=50; CoefficientList[Series[Product[(1+q^m)^32,{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)^32)) \\ G. C. Greubel, Mar 20 2018

Formula

a(n) ~ exp(Pi * 4 * sqrt(2*n/3)) / (65536 * 6^(1/4) * n^(3/4)). - Vaclav Kotesovec, Mar 05 2015

Extensions

Terms a(19) onward added by G. C. Greubel, Mar 20 2018