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%I #41 Apr 18 2017 07:04:12
%S 0,0,2,12,74,540,4682,47292,545834,7087260,102247562,1622632572,
%T 28091567594,526858348380,10641342970442,230283190977852,
%U 5315654681981354,130370767029135900,3385534663256845322,92801587319328411132,2677687796244384203114,81124824998504073881820
%N E.g.f.: (exp(x)-1)^2/(2-exp(x)).
%C Previous name was: A simple grammar.
%C Stirling transform of A005359(n-1)=[0,0,2,0,24,0,...] is a(n-1)=[0,0,2,12,74,...]. - _Michael Somos_, Mar 04 2004
%C Stirling transform of -(-1)^n*A052566(n-1)=[1,-1,4,-6,48,...] is a(n-1)=[1,0,2,12,74,...]. - _Michael Somos_, Mar 04 2004
%C Stirling transform of A000142(n)=[0,2,6,24,120,...] is a(n)=[0,2,12,74,...]. - _Michael Somos_, Mar 04 2004
%C Stirling transform of A007680(n)=[2,10,42,216,...] is a(n+1)=[2,12,74,...]. - _Michael Somos_, Mar 04 2004
%C a(n) is the number of chains in the power set of {1,2,...,n} that do not contain the empty set and do not contain {1,2,...,n}. Equivalently, a(n) is the number of ordered set partitions of {1,2,...,n} into at least 2 classes. - _Geoffrey Critzer_, Sep 01 2014
%H Vincenzo Librandi, <a href="/A052875/b052875.txt">Table of n, a(n) for n = 0..200</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=846">Encyclopedia of Combinatorial Structures 846</a>
%H R. B. Nelsen and H. Schmidt, Jr., <a href="http://www.jstor.org/stable/2690450">Chains in power sets</a>, Math. Mag., 64 (1991), 23-31.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Ordered_Bell_number">Ordered Bell number</a>
%F Second column of A084416: Sum_{i=2..n} i!*Stirling2(n, i) = A000670(n)-1. - _Vladeta Jovovic_, Sep 15 2003
%F E.g.f.: (exp(x)-1)^2/(2-exp(x)).
%F a(n) ~ n! / (2 * (log(2))^(n+1)). - _Vaclav Kotesovec_, Feb 25 2014
%F E.g.f.: A(x)*(1/(1 - A(x)) - 1) where A(x)=exp(x)-1. - _Geoffrey Critzer_, Sep 01 2014
%e a(3) = 12 because we have: {{1}}, {{2}}, {{3}}, {{1,2}}, {{1,3}}, {{2,3}}, {{1}, {1,2}}, {{1}, {1,3}}, {{2}, {1,2}}, {{2}, {2,3}}, {{3}, {1,3}}, {{3}, {2,3}}. - _Geoffrey Critzer_, Sep 01 2014
%p spec := [S, {B = Set(Z, 1 <= card), C = Sequence(B, 1 <= card), S=Prod(B, C)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
%t CoefficientList[Series[(E^x-1)^2/(2-E^x), {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Feb 25 2014 *)
%o (PARI) a(n)=if(n<0,0,n!*polcoeff(subst(y^2/(1-y),y,exp(x+x*O(x^n))-1),n))
%o (Sage)
%o def A052875(n):
%o return add(add((-1)^(j-i)*binomial(j,i)*i^n for i in range(n+1)) for j in range(n+1)) - 1
%o [A052875(n) for n in range(19)] # _Peter Luschny_, Jul 22 2014
%Y Cf. A007047, A038719.
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E New name using e.g.f., _Vaclav Kotesovec_, Feb 25 2014
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