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%I #78 Sep 08 2022 08:45:55
%S 1,2,4,10,30,104,406,1754,8280,42294,231950,1357140,8427194,55288874,
%T 381798644,2765917090,20960284294,165729739608,1364153612318,
%U 11665484410114,103448316470744,949739632313502,9013431476894646,88304011710168692,891917738589610578,9277180664459998706
%N a(n) = Bell(n)*(2 - 0^n).
%C Row sums of number triangle A186020.
%C a(n) is the number of collections of subsets of {1,2,...,n-1} that are pairwise disjoint. a(n+1) = 2*Sum_{j=0..n} C(n,j)*Bell(j). For example a(3)=10 because we have: {}, {{}}, {{1}}, {{2}}, {{1,2}}, {{},{1}}, {{},{2}}, {{},{1,2}}, {{1},{2}}, {{},{1},{2}}. - _Geoffrey Critzer_, Aug 28 2014
%H G. C. Greubel, <a href="/A186021/b186021.txt">Table of n, a(n) for n = 0..575</a>
%F E.g.f.: 2*exp(exp(x)-1)-1. - _Paul Barry_, Apr 06 2011
%F a(n) = A000110(n)*A040000(n).
%F a(n+1) = 1 + Sum_{k=0..n} C(n,k)*a(k). - _Franklin T. Adams-Watters_, Oct 02 2011.
%F From _Sergei N. Gladkovskii_, Nov 11 2012 to Mar 29 2013: (Start) Continued fractions:
%F G.f.: A(x)= 1 + 2*x/(G(0)-x) where G(k)= 1 - x*(k+1)/(1 - x/G(k+1)).
%F G.f.: G(0)-1 where G(k) = 1-(x*k+1)/(x*k - 1 - x*(x*k - 1)/(x + (x*k + 1)/G(k+1))).
%F G.f.: (G(0)-2)/x - 1 where G(k) = 1 + 1/(1 - x/(x + (1 - x*k)/G(k+1))).
%F G.f.: (S-2)/x - 1 where S = 2*Sum_{k>=0} x^k/Product_{n=0..k-1}(1 - n*x).
%F G.f.: 1/(1-x) - x/(G(0)-x^2+x) where G(k) =x^2 + x - 1 + k*(2*x-x^2) - x^2*k^2 + x*(x*k - 1)*(x*k + 2*x - 1)^2/G(k+1).
%F E.g.f.: E(0) - 1 where E(k) = 1 + 1/(1 - 1/(1 + (k+1)/x*bell(k)/bell(k+1)/E(k+1))). (End)
%F a(n) = A060719(n-1) + 1, and the inverse binomial transform of A060719. - _Gary W. Adamson_, May 20 2013
%F G.f. A(x) satisfies: A(x) = 1 + (x/(1 - x)) * (1 + A(x/(1 - x))). - _Ilya Gutkovskiy_, Jun 30 2020
%e a(4) = A060719(3) + 1 = 29 + 1 = 30.
%p A186021List := proc(m) local A, P, n; A := [1,2]; P := [2];
%p for n from 1 to m - 2 do P := ListTools:-PartialSums([P[-1], op(P)]);
%p A := [op(A), P[-1]] od; A end: A186021List(26); # _Peter Luschny_, Mar 24 2022
%t Prepend[Table[2 Sum[Binomial[n, j] BellB[j], {j, 0, n}], {n, 0, 25}], 1] (* _Geoffrey Critzer_, Aug 28 2014 *)
%t With[{nmax = 50}, CoefficientList[Series[2*Exp[Exp[x] - 1] - 1, {x, 0, nmax}], x]*Range[0, nmax]!] (* _G. C. Greubel_, Jul 24 2017 *)
%o (Magma) [Bell(n)*(2-0^n): n in [0..50]]; // _Vincenzo Librandi_, Apr 06 2011
%o (Python)
%o from itertools import accumulate
%o def A186021_list(size):
%o if size < 1: return []
%o L, accu = [1], [2]
%o for _ in range(size-1):
%o accu = list(accumulate([accu[-1]] + accu))
%o L.append(accu[0])
%o return L
%o print(A186021_list(26)) # _Peter Luschny_, Apr 25 2016
%o (PARI) x='x+O('x^50); Vec(serlaplace(2*exp(exp(x) - 1) -1)) \\ _G. C. Greubel_, Jul 24 2017
%Y Main diagonal of A271466 (shifted).
%K nonn,easy
%O 0,2
%A _Paul Barry_, Feb 10 2011
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