A014322 Convolution of Bell numbers with themselves.

1, 2, 5, 14, 44, 154, 595, 2518, 11591, 57672, 308368, 1762500, 10716321, 69011130, 468856113, 3348695194, 25064539520, 196052415230, 1598543907843, 13556379105766, 119332020447219, 1088376385244908, 10268343703117892, 100063762955374568, 1005822726810785809

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Equals row sums of triangle A144155. - Gary W. Adamson, Sep 12 2008

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012

Crossrefs

Cf. A000110, A014323, A014325, A144155.

Column k=2 of A292870.

Programs

Magma

A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
[A014322(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023

Maple

with(combinat):
a:= n-> add(bell(i)*bell(n-i), i=0..n):
seq(a(n), n=0..30);  # Alois P. Heinz, May 13 2014

Mathematica

a[n_]:= Sum[BellB[k]*BellB[n-k], {k,0,n}];
Table[a[n], {n,0,30}] (* Jean-François Alcover, Jan 17 2016 *)

SageMath

def A014322(n): return sum(bell_number(j)*bell_number(n-j) for j in range(n+1))
[A014322(n) for n in range(41)] # G. C. Greubel, Jan 08 2023

Formula

G.f.: (1/(1 - x - x^2/(1 - 2*x - 2*x^2/(1 - 3*x - 3*x^2/(1 - 4*x - 4*x^2/(1 - ...))))))^2, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017

G.f.: ( Sum_{j>=0} A000110(j)*x^j )^2. - G. C. Greubel, Jan 08 2023