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

0,2

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

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A000110, A014323, A014325, A144155.

Column k=2 of A292870.

Sequence in context: A081558 A202059 A350492 * A095148 A368636 A060996

Adjacent sequences: A014319 A014320 A014321 * A014323 A014324 A014325

Keyword

nonn

Author

N. J. A. Sloane

Status

approved