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A307755 Exponential convolution of partition numbers (A000041) with themselves.

%I A307755 #12 May 06 2019 10:00:17
%S A307755 1,2,6,18,58,184,586,1822,5618,16980,50892,150064,439210,1268924,
%T A307755 3640342,10337596,29160638,81570368,226795202,626070664,1718783084,
%U A307755 4689582366,12730998988,34373603158,92385339242,247099560046,658137847408,1745322097886,4610549234836,12131656526628
%N A307755 Exponential convolution of partition numbers (A000041) with themselves.
%H A307755 Vaclav Kotesovec, <a href="/A307755/b307755.txt">Table of n, a(n) for n = 0..3000</a>
%F A307755 E.g.f.: (Sum_{k>=0} A000041(k)*x^k/k!)^2.
%F A307755 a(n) = Sum_{k=0..n} binomial(n,k)*A000041(k)*A000041(n-k).
%F A307755 a(n) ~ exp(2*Pi*sqrt(n/3)) * 2^(n-2) / (3*n^2). - _Vaclav Kotesovec_, May 06 2019
%p A307755 a:= n-> (p-> add(binomial(n, j)*p(j)*p(n-j), j=0..n))(combinat[numbpart]):
%p A307755 seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 26 2019
%t A307755 nmax = 29; CoefficientList[Series[Sum[PartitionsP[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!
%t A307755 Table[Sum[Binomial[n, k] PartitionsP[k] PartitionsP[n - k], {k, 0, n}], {n, 0, 29}]
%Y A307755 Cf. A000041, A000712, A218481, A307756.
%K A307755 nonn
%O A307755 0,2
%A A307755 _Ilya Gutkovskiy_, Apr 26 2019