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

0,2

Links

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

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

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}]

Crossrefs

Cf. A000009, A022567, A266232, A307755.

Sequence in context: A090032 A090377 A151278 * A149810 A095337 A162533

Adjacent sequences: A307753 A307754 A307755 * A307757 A307758 A307759

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 26 2019

Status

approved