A307756 Exponential convolution of number of partitions into distinct parts (A000009) with themselves.

1, 2, 4, 10, 26, 66, 184, 472, 1268, 3340, 8748, 22772, 59102, 151590, 386830, 983914, 2489384, 6263284, 15703204, 39221884, 97498736, 241538472, 596115898, 1465958522, 3595196600, 8788765304, 21421616934, 52080152238, 126268822824, 305365334180, 736770528064

Offset: 0

Author

Ilya Gutkovskiy, Apr 26 2019

Keywords

• nonn

Links

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

Crossrefs

Cf. A000009, A022567, A266232, A307755.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= n-> add(binomial(n, j)*b(j)*b(n-j), j=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 26 2019

Mathematica

nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] PartitionsQ[k] PartitionsQ[n - k], {k, 0, n}], {n, 0, 30}]

Formula

E.g.f.: (Sum_{k>=0} A000009(k)*x^k/k!)^2.

a(n) = Sum_{k=0..n} binomial(n,k)*A000009(k)*A000009(n-k).

a(n) ~ exp(Pi*sqrt(2*n/3)) * 2^(n - 5/2) / (sqrt(3) * n^(3/2)). - Vaclav Kotesovec, May 06 2019