A070906 Every third Bell number A000110.

1, 5, 203, 21147, 4213597, 1382958545, 682076806159, 474869816156751, 445958869294805289, 545717047936059989389, 846749014511809332450147, 1629595892846007606764728147, 3819714729894818339975525681317

Offset: 0

Author

Benoit Cloitre, May 19 2002

Keywords

• easy
• nonn

Crossrefs

Cf. A000110, A020557.

Programs

Mathematica

Table[ BellB[3*n], {n, 0, 12}] (* Jean-François Alcover, Dec 13 2012 *)
BellB[3*Range[0,15]] (* Harvey P. Dale, Apr 19 2020 *)

PARI

for(n=0,50,print1(round(sum(i=0,1000,i^(3*n)/(i)!)/exp(1)),","))

Python

from itertools import accumulate, islice
def A070906_gen(): # generator of terms
    yield 1
    blist, b = (1,), 1
    while True:
        for _ in range(3):
            blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield b
A070906_list = list(islice(A070906_gen(),30)) # Chai Wah Wu, Jun 22 2022

Sage

[bell_number(3*n) for n in range(0, 13)] # Zerinvary Lajos, May 14 2009

Formula

a(n) = Bell(3*n) = A000110(3*n). - Vladeta Jovovic, Feb 02 2003

a(n) = exp(-1)*Sum_{k>=0} k^(3n)/k!.

E.g.f.: exp(x*(d_z)^3)*(exp(exp(z)-1)) |_{z=0}, with the derivative operator d_z := d/dz. Adapted from eqs. (14) and (15) of the 1999 C. M. Bender reference given in A000110.

E.g.f.: exp(-1)*Sum_{n>=0} exp(n^3*x)/n!. - Vladeta Jovovic, Aug 24 2006