A014325 Four-fold convolution of Bell numbers with themselves.

1, 4, 14, 48, 169, 624, 2442, 10188, 45452, 217100, 1109914, 6064584, 35330715, 218788432, 1435302930, 9940062428, 72422364227, 553338786504, 4420324121772, 36820875272488, 319053830821880, 2869645346679368, 26739383194844404, 257682847299543248

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..575

Crossrefs

Cf. A000110, A014322, A014323.

Column k=4 of A292870.

Programs

Magma

A014322:= func< n | (&+[Bell(j)*Bell(n-j): j in [0..n]]) >;
A014325:= func< n | (&+[A014322(j)*A014322(n-j): j in [0..n]]) >;
[A014325(n): n in [0..40]]; // G. C. Greubel, Jan 08 2023

Mathematica

A014322[n_]:= Sum[BellB[j]*BellB[n-j], {j,0,n}];
A014325[n_]:= Sum[A014322[j]*A014322[n-j], {j,0,n}];
Table[A014325[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 A014325(n): return sum(A014322(j)*A014322(n-j) for j in range(n+1))
[A014325(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 - ...))))))^4, a continued fraction. - Ilya Gutkovskiy, Sep 25 2017

G.f.: ( Sum_{j>=0} A000110(j)*x^j )^4. - G. C. Greubel, Jan 08 2023