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A264686 Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - x^k).

%I A264686 #19 Aug 28 2019 10:04:36
%S A264686 1,3,6,15,27,51,93,159,264,432,696,1086,1683,2553,3837,5700,8367,
%T A264686 12147,17505,24972,35361,49728,69402,96243,132657,181782,247692,
%U A264686 335838,453042,608289,813102,1082256,1434519,1894215,2491644,3265869,4265973,5553771,7207167
%N A264686 Expansion of Product_{k>=1} (1 + 2*x^k)/(1 - x^k).
%C A264686 Convolution of A000041 and A032302.
%H A264686 Vaclav Kotesovec, <a href="/A264686/b264686.txt">Table of n, a(n) for n = 0..10000</a>
%F A264686 a(n) ~ sqrt(c) * exp(sqrt(2*c*n)) / (4*Pi*sqrt(3)*n), where c = 2*Pi^2/3 + log(2)^2 + 2*polylog(2, -1/2) = 6.163360867463814765670634381079217086937812673723341... . - _Vaclav Kotesovec_, Jan 04 2016
%p A264686 b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
%p A264686       `if`(n=0, 1, b(n, i-1)+`if`(i>n, 0, 2*b(n-i, i-1))))
%p A264686     end:
%p A264686 a:= n-> add(b(i$2)*combinat[numbpart](n-i), i=0..n):
%p A264686 seq(a(n), n=0..60);  # _Alois P. Heinz_, Dec 22 2017
%t A264686 nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%o A264686 (PARI) { my(n=40); Vec(prod(k=1, n, 3/(1-x^k) - 2 + O(x*x^n))) } \\ _Andrew Howroyd_, Dec 22 2017
%Y A264686 Cf. A000041, A006951, A015128, A032302, A261584, A264685, A266821.
%Y A264686 Column k=3 of A321884.
%K A264686 nonn
%O A264686 0,2
%A A264686 _Vaclav Kotesovec_, Nov 21 2015