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A261561 Expansion of Product_{k>=1} (1/(1 - 2*x^k))^k.

%I A261561 #17 Sep 21 2018 11:27:49
%S A261561 1,2,8,22,64,162,424,1022,2480,5770,13336,30046,67184,147554,321592,
%T A261561 692278,1479568,3133474,6596008,13788606,28679264,59335530,122256456,
%U A261561 250875550,513116864,1046190786,2127557592,4316282006,8739096992,17661731138,35639764536
%N A261561 Expansion of Product_{k>=1} (1/(1 - 2*x^k))^k.
%H A261561 Alois P. Heinz, <a href="/A261561/b261561.txt">Table of n, a(n) for n = 0..2000</a> (first 1001 terms from Vaclav Kotesovec)
%F A261561 a(n) ~ c * 2^n, where c = Product_{j>=1} 1/(1 - 1/2^j)^(j+1) = 34.7387234654851595844514193757064296508992247003230539635669599773458896...
%F A261561 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} 2^d * n^2/d^2 ). - _Paul D. Hanna_, Sep 30 2015
%p A261561 b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,
%p A261561       add(2^j*binomial(i+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p A261561     end:
%p A261561 a:= n-> b(n$2):
%p A261561 seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 21 2018
%t A261561 nmax = 50; CoefficientList[Series[Product[(1/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]
%t A261561 nmax = 50; CoefficientList[Series[Exp[Sum[2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]
%o A261561 (PARI) {a(n) = polcoeff( exp( sum(m=1,n,x^m/m * sumdiv(m,d,2^d*m^2/d^2) ) +x*O(x^n)),n)}
%o A261561 for(n=0,40,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 30 2015
%Y A261561 Cf. A000219, A070933, A246935, A261562, A261563, A261565.
%K A261561 nonn
%O A261561 0,2
%A A261561 _Vaclav Kotesovec_, Aug 24 2015