A022726 Expansion of 1/Product_{m>=1} (1 - m*q^m)^2.

1, 2, 7, 18, 49, 114, 282, 624, 1422, 3058, 6597, 13700, 28564, 57698, 116479, 230398, 453698, 879080, 1696732, 3230578, 6124326, 11486884, 21439480, 39659598, 73036175, 133445640, 242756058, 438680734, 789328034, 1411926186, 2515574329, 4458203590, 7871211452, 13831782146

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Self-convolution of A006906. - Vaclav Kotesovec, Jan 06 2016

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..6000

Crossrefs

Cf. A006906, A022662.

Column k=2 of A297328.

Programs

Magma

n:=40; R:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^2:m in [1..n]])); // G. C. Greubel, Jul 25 2018

Mathematica

nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 06 2016 *)

PARI

N=66; q='q+O('q^N);
gf= 1/prod(n=1,N, (1-n*q^n)^2 );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

Formula

From Vaclav Kotesovec, Jan 07 2016: (Start)

a(n) ~ c * n * 3^(n/3), where

c = 9588921272.54120308291761424720457... = (c0^2 + 2*c1*c2)/3 if mod(n,3)=0

c = 9588921272.50566179874517327053929... = (c2^2 + 2*c0*c1)/3 if mod(n,3)=1

c = 9588921272.49785814355801212400055... = (c1^2 + 2*c0*c2)/3 if mod(n,3)=2

For the constants c0, c1, c2 see A006906.

(End)

G.f.: exp(2*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - Ilya Gutkovskiy, Feb 07 2018

Extensions

Added more terms, Joerg Arndt, Oct 06 2012