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%I #7 Sep 15 2021 10:25:46
%S 1,3,8,27,67,189,509,1329,3344,8694,22062,54756,136741,335103,822277,
%T 2016738,4872787,11711655,28253743,67319328,160333627,381350646,
%U 901272326,2121969771,4991176893,11689645776,27305992220,63705989106,148106539514,343371565449,795524336390
%N a(n) = Sum_{k=0..n} 2^k * A000009(k) * A000041(n-k).
%F a(n) ~ A065446 * 2^n * A000009(n).
%F a(n) ~ 2^(n-2) * exp(Pi*sqrt(n/3)) / (3^(1/4) * QPochhammer(1/2) * n^(3/4)).
%F G.f.: Product_{k>=1} (1 + 2^k*x^k) / (1 - x^k).
%t Table[Sum[2^k*PartitionsQ[k]*PartitionsP[n-k], {k, 0, n}], {n, 0, 50}]
%t nmax = 50; CoefficientList[Series[Product[(1 + 2^k*x^k) / (1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%Y Cf. A000009, A000041, A015128, A065446, A264686, A347829.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Sep 15 2021
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