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

%I A022726 #25 Sep 08 2022 08:44:46
%S A022726 1,2,7,18,49,114,282,624,1422,3058,6597,13700,28564,57698,116479,
%T A022726 230398,453698,879080,1696732,3230578,6124326,11486884,21439480,
%U A022726 39659598,73036175,133445640,242756058,438680734,789328034,1411926186,2515574329,4458203590,7871211452,13831782146
%N A022726 Expansion of 1/Product_{m>=1} (1 - m*q^m)^2.
%C A022726 Self-convolution of A006906. - _Vaclav Kotesovec_, Jan 06 2016
%H A022726 Vaclav Kotesovec, <a href="/A022726/b022726.txt">Table of n, a(n) for n = 0..6000</a>
%F A022726 From _Vaclav Kotesovec_, Jan 07 2016: (Start)
%F A022726 a(n) ~ c * n * 3^(n/3), where
%F A022726 c = 9588921272.54120308291761424720457... = (c0^2 + 2*c1*c2)/3  if mod(n,3)=0
%F A022726 c = 9588921272.50566179874517327053929... = (c2^2 + 2*c0*c1)/3  if mod(n,3)=1
%F A022726 c = 9588921272.49785814355801212400055... = (c1^2 + 2*c0*c2)/3  if mod(n,3)=2
%F A022726 For the constants c0, c1, c2 see A006906.
%F A022726 (End)
%F A022726 G.f.: exp(2*Sum_{j>=1} Sum_{k>=1} k^j*x^(j*k)/j). - _Ilya Gutkovskiy_, Feb 07 2018
%t A022726 nmax = 40; CoefficientList[Series[Product[1/(1-k*x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jan 06 2016 *)
%o A022726 (PARI)
%o A022726 N=66; q='q+O('q^N);
%o A022726 gf= 1/prod(n=1,N, (1-n*q^n)^2 );
%o A022726 Vec(gf)
%o A022726 /* _Joerg Arndt_, Oct 06 2012 */
%o A022726 (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 A022726 Cf. A006906, A022662.
%Y A022726 Column k=2 of A297328.
%K A022726 nonn
%O A022726 0,2
%A A022726 _N. J. A. Sloane_
%E A022726 Added more terms, _Joerg Arndt_, Oct 06 2012