A247452 a(n) = 3^n*Bell(n).

1, 3, 18, 135, 1215, 12636, 147987, 1917999, 27162540, 416236401, 6848207775, 120206639790, 2239278203277, 44074161731151, 913065539247018, 19843943547060315, 451135755042249987, 10701182793462338052, 264250529777677991751, 6779171511882363638619, 180350988089950776032172

Offset: 0

Author

Karol A. Penson, Sep 17 2014

Keywords

• nonn

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000110, A055882.

Programs

Magma

[3^n*Bell(n): n in [0..20]]; // Vincenzo Librandi, Sep 19 2014

Mathematica

Table[3^n BellB[n], {n, 0, 20}] (* Vincenzo Librandi, Sep 19 2014 *)

Python

# Python 3.2 or above required.
from itertools import accumulate
A247452_list, blist, b, n3 = [1,3], [1], 1, 9
for _ in range(2,201):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A247452_list.append(b*n3)
    n3 *= 3 # Chai Wah Wu, Sep 19 2014

Formula

a(n) = 3^n*A000110(n).

a(n) = (1/e)*Sum_{k>=0} (3*k)^n/k!. (This is a Dobinski-type formula.)

O.g.f.: (1/e)*Sum_{k>=0} 1/(k!*(1-3*k*z)).

E.g.f.: exp(exp(3*z)-1).

a(n) is the n-th moment of a discrete, positive weight function w(x) consisting of an infinite comb of Dirac delta functions situated at x=3*k, with k = 0, 1, ..., defined as w(x) = (1/e)*Sum_{k>=0} Dirac(x-3*k)/k!.

G.f.: 1/(1-3*x/(1-3*x/(1-3*x/(1-6*x/(1-3*x/(1-9*x/(1-...))))))) (continued fraction). - Philippe Deléham, Sep 18 2014

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * 3^k * a(n-k). - Ilya Gutkovskiy, Jan 16 2020