A055882 - OEIS

%I #60 Jun 08 2024 15:44:11

%S 1,2,8,40,240,1664,12992,112256,1059840,10827264,118758400,1389711360,

%T 17258893312,226463227904,3127694491648,45316785602560,

%U 686826595745792,10861264214949888,178802342273744896,3058036745204924416,54236710945813430272,995874184692762673152

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

%C 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

%H Chai Wah Wu, <a href="/A055882/b055882.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = exp(-1)*2^n*sum(k=>0, k^n/k!). - _Benoit Cloitre_, May 20 2002

%F G.f.: 1/(1-2x/(1-2x/(1-2x/(1-4x/(1-2x/(1-6x/(1-2x/(1-8x/(1-... (continued fraction). [_Paul Barry_, Oct 11 2009]

%F 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

%F 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

%F 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

%F 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

%F a(n) = Sum_{k=0..n} binomial(n,k) * A004211(k) * A004211(n-k). - _Vaclav Kotesovec_, Apr 17 2020

%p seq(add(binomial(n, k)*(bell(n)), k=0..n), n=0..18); # _Zerinvary Lajos_, Dec 01 2006

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p       a(n-j) *binomial(n-1, j-1)*2^j, j=1..n))

%p     end:

%p seq(a(n), n=0..23);  # _Alois P. Heinz_, Oct 04 2019

%t nn=20;a=Exp[2x]-1;Range[0,nn]!CoefficientList[Series[Exp[a],{x,0,nn}],x]  (* _Geoffrey Critzer_, Sep 16 2012 *)

%t Table[2^n BellB[n], {n, 0, 20}] (* _Vincenzo Librandi_, Sep 19 2014 *)

%o (Python)

%o # Python 3.2 or higher required

%o from itertools import accumulate

%o A055882_list, blist, b, n2 = [1,2], [1], 1, 4

%o for _ in range(2, 201):

%o     blist = list(accumulate([b]+blist))

%o     b = blist[-1]

%o     A055882_list.append(b*n2)

%o     n2 *= 2 # Chai Wah Wu, Sep 19 2014

%o (Magma) [2^n*Bell(n): n in [0..20]]; // _Vincenzo Librandi_, Sep 19 2014

%Y Cf. A000079, A000110, A055883, A143405.

%K nonn

%O 0,2

%A _Christian G. Bower_, Jun 09 2000