A022726 - OEIS

%I #25 Sep 08 2022 08:44:46

%S 1,2,7,18,49,114,282,624,1422,3058,6597,13700,28564,57698,116479,

%T 230398,453698,879080,1696732,3230578,6124326,11486884,21439480,

%U 39659598,73036175,133445640,242756058,438680734,789328034,1411926186,2515574329,4458203590,7871211452,13831782146

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

%C Self-convolution of A006906. - _Vaclav Kotesovec_, Jan 06 2016

%H Vaclav Kotesovec, <a href="/A022726/b022726.txt">Table of n, a(n) for n = 0..6000</a>

%F From _Vaclav Kotesovec_, Jan 07 2016: (Start)

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

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

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

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

%F For the constants c0, c1, c2 see A006906.

%F (End)

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

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

%o (PARI)

%o N=66; q='q+O('q^N);

%o gf= 1/prod(n=1,N, (1-n*q^n)^2 );

%o Vec(gf)

%o /* _Joerg Arndt_, Oct 06 2012 */

%o (Magma) n:=40; R<x>:=PowerSeriesRing(Integers(), n); Coefficients(R!(&*[(1/(1-m*x^m))^2:m in [1..n]])); // _G. C. Greubel_, Jul 25 2018

%Y Cf. A006906, A022662.

%Y Column k=2 of A297328.

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E Added more terms, _Joerg Arndt_, Oct 06 2012