A307755 - OEIS

%I #12 May 06 2019 10:00:17

%S 1,2,6,18,58,184,586,1822,5618,16980,50892,150064,439210,1268924,

%T 3640342,10337596,29160638,81570368,226795202,626070664,1718783084,

%U 4689582366,12730998988,34373603158,92385339242,247099560046,658137847408,1745322097886,4610549234836,12131656526628

%N Exponential convolution of partition numbers (A000041) with themselves.

%H Vaclav Kotesovec, <a href="/A307755/b307755.txt">Table of n, a(n) for n = 0..3000</a>

%F E.g.f.: (Sum_{k>=0} A000041(k)*x^k/k!)^2.

%F a(n) = Sum_{k=0..n} binomial(n,k)*A000041(k)*A000041(n-k).

%F a(n) ~ exp(2*Pi*sqrt(n/3)) * 2^(n-2) / (3*n^2). - _Vaclav Kotesovec_, May 06 2019

%p a:= n-> (p-> add(binomial(n, j)*p(j)*p(n-j), j=0..n))(combinat[numbpart]):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Apr 26 2019

%t nmax = 29; CoefficientList[Series[Sum[PartitionsP[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[Binomial[n, k] PartitionsP[k] PartitionsP[n - k], {k, 0, n}], {n, 0, 29}]

%Y Cf. A000041, A000712, A218481, A307756.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 26 2019