A055882 a(n) = 2^n*Bell(n). E.g.f.: exp(exp(2*x)-1).

1, 2, 8, 40, 240, 1664, 12992, 112256, 1059840, 10827264, 118758400, 1389711360, 17258893312, 226463227904, 3127694491648, 45316785602560, 686826595745792, 10861264214949888, 178802342273744896, 3058036745204924416, 54236710945813430272, 995874184692762673152

Offset: 0

Author

Christian G. Bower, Jun 09 2000

Keywords

• nonn
• changed

Comments

a(n) is the number of set partitions of {1,2,...,n} with a (possibly empty) subset of designated elements in each block. - Geoffrey Critzer, Sep 16 2012

Links

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

Crossrefs

Cf. A000079, A000110, A055883, A143405.

Programs

Magma

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

Maple

seq(add(binomial(n, k)*(bell(n)), k=0..n), n=0..18); # Zerinvary Lajos, Dec 01 2006
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j) *binomial(n-1, j-1)*2^j, j=1..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 04 2019

Mathematica

nn=20;a=Exp[2x]-1;Range[0,nn]!CoefficientList[Series[Exp[a],{x,0,nn}],x]  (* Geoffrey Critzer, Sep 16 2012 *)
Table[2^n BellB[n], {n, 0, 20}] (* Vincenzo Librandi, Sep 19 2014 *)

Python

# Python 3.2 or higher required
from itertools import accumulate
A055882_list, blist, b, n2 = [1,2], [1], 1, 4
for _ in range(2, 201):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A055882_list.append(b*n2)
    n2 *= 2 # Chai Wah Wu, Sep 19 2014

Formula

a(n) = exp(-1)*2^n*Sum_{k>=0} k^n/k!. - Benoit Cloitre, May 20 2002

G.f.: 1/(1-2*x/(1-2*x/(1-2*x/(1-4*x/(1-2*x/(1-6*x/(1-2*x/(1-8*x/(1-... (continued fraction). - Paul Barry, Oct 11 2009

G.f.: 1/(U(0) - 2*x) where U(k) = 1 + 2*x - 2*x*(k+1)/(1 - 2*x/U(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Oct 12 2012

G.f.: G(0)/(1+2*x) where G(k) = 1 - 4*x*(k+1)/((2*k+1)*(4*x*k-1) - 2*x*(2*k+1)*(2*k+3)*(4*x*k-1)/(2*x*(2*k+3) - 2*(k+1)*(4*x*k+2*x-1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2012

G.f.: G(0)/2 where G(k) = 1 - (2*x*k + 1)/(2*x*k - 1 - 2*x*(2*x*k - 1)/(2*x + (2*x*k + 1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 30 2013

G.f.: 1/Q(0), where Q(k) = 1 - 2*(k+1)*x - 4*(k+1)*x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013

a(n) = Sum_{k=0..n} binomial(n,k) * A004211(k) * A004211(n-k). - Vaclav Kotesovec, Apr 17 2020