A014323 Three-fold convolution of Bell numbers with themselves.

1, 3, 9, 28, 93, 333, 1289, 5394, 24366, 118526, 618924, 3456942, 20573391, 129951231, 867877107, 6106194478, 45109290477, 348836705235, 2816093142803, 23673989688810, 206794355179656, 1873232870155036, 17565534522745008, 170237112831874188

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576

Crossrefs

Cf. A000110, A014322, A014325.

Column k=3 of A292870.

Programs

Magma

A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
A014323:= func< n | (&+[Bell(j)*A014322(n-j): j in [0..n]]) >;
[A014323(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023

Mathematica

A014322[n_]:= Sum[BellB[j]*BellB[n-j], {j,0,n}];
A014323[n_]:= Sum[BellB[j]*A014322[n-j], {j,0,n}];
Table[A014323[n], {n,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))
def A014323(n): return sum(bell_number(j)*A014322(n-j) for j in range(n+1))
[A014323(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 - ...))))))^3, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017

From G. C. Greubel, Jan 08 2023: (Start)

a(n) = Sum_{j=0..n} A000110(j)*A014322(n-j).

G.f.: ( Sum_{j>=0} A000110(j)*x^j )^3. (End)