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A104533 E.g.f.: exp(2x/(1-2x)).

%I A104533 #35 Nov 10 2017 03:12:18
%S A104533 1,2,12,104,1168,16032,259264,4817024,100954368,2353435136,
%T A104533 60355677184,1687701792768,51077784506368,1662782678736896,
%U A104533 57917727119818752,2148722382829027328,84569896954751942656,3518839711497761980416,154306731918073225019392
%N A104533 E.g.f.: exp(2x/(1-2x)).
%C A104533 Number of hierarchical orderings for n labeled elements (see A075729) when there are two kinds A and B of elements.
%H A104533 Seiichi Manyama, <a href="/A104533/b104533.txt">Table of n, a(n) for n = 0..398</a> (terms 0..150 from Alois P. Heinz)
%H A104533 N. J. A. Sloane and Thomas Wieder, <a href="https://arxiv.org/abs/math/0307064">The Number of Hierarchical Orderings</a>, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
%F A104533 a(n) = 2^n*A000262(n) = 2^n*n!*Sum_{k=0..n} C(n-1,k)/(k+1)!. - _Paul Barry_, Apr 28 2007
%F A104533 With p(n) = the number of integer partitions of n, d(i) = the number of different parts of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, sum_{i=1}^{p(n)} = sum over i and prod_{j=1}^{d(i)} = product over j one has: a(n)=sum_{i=1}^{p(n)} n!/(prod_{j=1}^{d(i)} m(i, j)!) * 2^(n)
%F A104533 E.g.f.: E(0)/2, where E(k)= 1 + 1/(1 - 2*x/(2*x + (k+1)*(1-2*x)/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 09 2013
%F A104533 E.g.f.: E(0) - 1, where E(k) = 2 + 2*x/((2*k+1)*(1-2*x) - 2*x/E(k+1) ); (continued fraction ). - _Sergei N. Gladkovskii_, Dec 31 2013
%e A104533 Let "a_i" and "b_j" be elements situated in the classes A and B with _i and _j as labels. Let : denote a separator among levels (ranks). Let | denote a separator among groups. E.g., a_1:b_2|b_1 is a hierarchy composed of two groups which contain three elements in total.
%e A104533 a(2) = 12 from b_2:b_1, b_2:a_1, b_2|b_1, a_1:a_2, b_2:a_1, a_1|a_2, a_1:b_2, a_2:a_1, b_1:a_2, a_2:b_1, b_1|a_2, b_2:b_1.
%p A104533 SetSeqUnnL := [T, {T=Set(S,card>=1), S=Sequence(U,card>=1), U=Union(a,b),a=Atom, b=Atom},labeled]; seq(count(SetSeqUnnL,size=j),j=1..20);
%p A104533 A104533 := proc(n::integer) local i,j,prttnlst,prttn,liste,ZahlVerschiedenerTeile,H,Mltplztt; Mltplztt:=vector[1000]; prttnlst:=partition(n); H := 0; for i from 1 to nops(prttnlst) do prttn := prttnlst[i]; liste := convert(prttn,multiset); ZahlVerschiedenerTeile := nops(liste); for j from 1 to ZahlVerschiedenerTeile do Mltplztt[j] := op(2,op(j,liste)); od; H := H + (n!/mul(Mltplztt[j]!,j=1..ZahlVerschiedenerTeile)) * 2^n; od; print(n,H); end proc;
%t A104533 CoefficientList[Exp[2 x/(1 - 2 x)] + O[x]^21, x]*Range[0, 20]!
%t A104533 (* or: *)
%t A104533 a[0] = 1; a[n_] := 2^n*n!*Hypergeometric1F1[n + 1, 2, 1]/E;
%t A104533 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Nov 10 2017 *)
%Y A104533 Cf. A075729, A034691, A034899.
%Y A104533 Equals 2^n * A000262(n).
%K A104533 nonn
%O A104533 0,2
%A A104533 _Thomas Wieder_, Mar 13 2005
%E A104533 Edited by _N. J. A. Sloane_, May 06 2008, at the suggestion of _Joerg Arndt_