This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A055882 #64 Sep 03 2025 05:02:59
%S A055882 1,2,8,40,240,1664,12992,112256,1059840,10827264,118758400,1389711360,
%T A055882 17258893312,226463227904,3127694491648,45316785602560,
%U A055882 686826595745792,10861264214949888,178802342273744896,3058036745204924416,54236710945813430272,995874184692762673152
%N A055882 a(n) = 2^n*Bell(n). E.g.f.: exp(exp(2*x)-1).
%C A055882 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 A055882 Chai Wah Wu, <a href="/A055882/b055882.txt">Table of n, a(n) for n = 0..200</a>
%F A055882 a(n) = exp(-1)*2^n*Sum_{k>=0} k^n/k!. - _Benoit Cloitre_, May 20 2002
%F A055882 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
%F A055882 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 A055882 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 A055882 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 A055882 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 A055882 a(n) = Sum_{k=0..n} binomial(n,k) * A004211(k) * A004211(n-k). - _Vaclav Kotesovec_, Apr 17 2020
%p A055882 seq(add(binomial(n, k)*(bell(n)), k=0..n), n=0..18); # _Zerinvary Lajos_, Dec 01 2006
%p A055882 # second Maple program:
%p A055882 a:= proc(n) option remember; `if`(n=0, 1, add(
%p A055882 a(n-j) *binomial(n-1, j-1)*2^j, j=1..n))
%p A055882 end:
%p A055882 seq(a(n), n=0..23); # _Alois P. Heinz_, Oct 04 2019
%t A055882 nn=20;a=Exp[2x]-1;Range[0,nn]!CoefficientList[Series[Exp[a],{x,0,nn}],x] (* _Geoffrey Critzer_, Sep 16 2012 *)
%t A055882 Table[2^n BellB[n], {n, 0, 20}] (* _Vincenzo Librandi_, Sep 19 2014 *)
%o A055882 (Python)
%o A055882 # Python 3.2 or higher required
%o A055882 from itertools import accumulate
%o A055882 A055882_list, blist, b, n2 = [1,2], [1], 1, 4
%o A055882 for _ in range(2, 201):
%o A055882 blist = list(accumulate([b]+blist))
%o A055882 b = blist[-1]
%o A055882 A055882_list.append(b*n2)
%o A055882 n2 *= 2 # _Chai Wah Wu_, Sep 19 2014
%o A055882 (Magma) [2^n*Bell(n): n in [0..20]]; // _Vincenzo Librandi_, Sep 19 2014
%Y A055882 Cf. A000079, A000110, A055883, A143405.
%K A055882 nonn,changed
%O A055882 0,2
%A A055882 _Christian G. Bower_, Jun 09 2000