A014326 Convolution of partition numbers and Bell numbers.

1, 2, 5, 12, 32, 95, 328, 1294, 5748, 28152, 149768, 856130, 5218107, 33712600, 229800588, 1646316230, 12355374717, 96861192976, 791258805462, 6720627186126, 59234364203973, 540812222400025, 5106663817693176, 49798678281859244, 500857393911224861

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000041, A000110.

Partial sums of A292503.

Programs

Magma

A014326:= func< n | (&+[NumberOfPartitions(j)*Bell(n-j): j in [0..n]]) >;
[A014326(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023

Maple

with(combinat):
a:= n-> add(numbpart(k)*bell(n-k), k=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 15 2015

Mathematica

a[n_]:= Sum[PartitionsP[k]*BellB[n-k], {k,0,n}];
Table[a[n], {n,0,30}] (* Jean-François Alcover, Dec 06 2016 *)

SageMath

def A014326(n): return sum(number_of_partitions(j)*bell_number(n-j) for j in range(n+1))
[A014326(n) for n in range(41)] # G. C. Greubel, Jan 08 2023