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

1, 2, 6, 18, 58, 184, 586, 1822, 5618, 16980, 50892, 150064, 439210, 1268924, 3640342, 10337596, 29160638, 81570368, 226795202, 626070664, 1718783084, 4689582366, 12730998988, 34373603158, 92385339242, 247099560046, 658137847408, 1745322097886, 4610549234836, 12131656526628

Offset: 0

Author

Ilya Gutkovskiy, Apr 26 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A000041, A000712, A218481, A307756.

Programs

Maple

a:= n-> (p-> add(binomial(n, j)*p(j)*p(n-j), j=0..n))(combinat[numbpart]):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 26 2019

Mathematica

nmax = 29; CoefficientList[Series[Sum[PartitionsP[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] PartitionsP[k] PartitionsP[n - k], {k, 0, n}], {n, 0, 29}]

Formula

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

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

a(n) ~ exp(2*Pi*sqrt(n/3)) * 2^(n-2) / (3*n^2). - Vaclav Kotesovec, May 06 2019