%I A083355
%S A083355 1,1,4,23,175,1662,18937,251729,3824282,65361237,1241218963,25928015368,
%T A083355 590852003947,14586471744301,387798817072596,11046531316503163,
%U A083355 335640299372252595,10835556229612637150,370383732831919278037
%N A083355 Number of preferential arrangements for the set partitions of the n-set 
               [1,2,3,...,n].
%C A083355 Labeled analogue of A055887. See combstruct commands for more precise 
               definition.
%C A083355 Stirling transform of A000670(n)=[1,3,13,75,...] is a(n)=[1,4,23,175,
               ...]. - Michael Somos Mar 04 2004
%D A083355 Thomas Wieder: The number of certain rankings and hierarchies formed 
               from labeled or unlabeled elements and sets, Applied Mathematical 
               Sciences, vol. 3, 2009, no. 55, 2707 - 2724. [From Thomas Wieder 
               (thomas.wieder(AT)t-online.de), Nov 14 2009]
%H A083355 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a 
               href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type 
               relations via substitution and the moment problem</a> [J. Phys. A 
               37 (2004), 3475-3487]
%H A083355 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A083355 N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/
               0307064">The Number of Hierarchical Orderings</a>, Order 21 (2004), 
               83-89.
%H A083355 Thomas Wieder, <a href="a083355.txt">Further comments on A083355</a>
%H A083355 Thomas Wieder, <a href="http://www.thomas-wieder.privat.t-online.de/default.html">
               Home Page</a>.
%H A083355 Thomas Wieder, <a href="http://homepages.tu-darmstadt.de/~wieder/Welcome.html">
               (Old) Home Page</a>.
%F A083355 E.g.f.: 1/(2-exp(exp(x)-1)).
%F A083355 Representation as a double infinite series (Dobinski-type formula), in 
               Maple notation: a(n)=sum(k^n/k!*sum(p^k/(2*exp(1))^p, p=1..infinity), 
               k=1..infinity)/2, n=1, 2... From Karol A. Penson (penson(AT)lptl.jussieu.fr) 
               and Pawel Blasiak (blasiak(AT)lptl.jussieu.fr), Nov 30 2003.
%F A083355 a(n) ~ n!/(2 * c * (log c)^(n+1)) where c = 1 + log 2.
%F A083355 a(n) = Sum_{k=1..n} binomial(n, k)*Bell(k)*a(n-k). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Jul 24 2003
%F A083355 a(n) = Sum_{i=1}^{n} Sum_{j=1}^{i} j!*stirling2(i, j)*stirling2(n, i) 
               - Thomas Wieder (wieder.thomas(AT)t-online.de), May 09 2005
%F A083355 a(n)=sum_{k=1}^n S2(n,k) A000670(k)
%e A083355 Let the colon ":" be a separator between two levels. E.g. in {1,2}:{3} 
               the set {1,2} is on the first level, the set {3} is on the second 
               level.
%e A083355 n=2 gives A083355(2)=4 because we have {1,2} {1}{2} {1}:{2} {2}:{1}.
%e A083355 n=3 gives A083355(3)=23 because we have:
%e A083355 {1,2,3}
%e A083355 {1,2}{3} {1,2}:{3} {3}:{1,2}
%e A083355 {1,3}{2} {1,3}:{2} {2}:{1,3}
%e A083355 {2,3}{1} {2,3}:{1} {1}:{2,3}
%e A083355 {1}{2}{3}
%e A083355 {1}:{2}:{3}
%e A083355 {3}:{1}:{2}
%e A083355 {2}:{3}:{1}
%e A083355 {1}:{3}:{2}
%e A083355 {2}:{1}:{3}
%e A083355 {3}:{2}:{1}
%e A083355 {1}{2}:{3} {1}{3}:{2} {2}{3}:{1}
%e A083355 {1}:{2}{3} {2}:{1}{3} {3}:{1}{2}.
%e A083355 Examples for the unlabeled case A055887:
%e A083355 n=2 gives A055887(2)=3 because {1,1} {{1}:{1}} {2}
%e A083355 n=3 gives A055887(3)=8 because {1,1,1} {{1}:{1,1}} {{1,1}:{1}} {{1}:{1}:{1}} 
               {1,2} {{1}:{2}} {{2}:{1}} {3}.
%p A083355 with(combstruct); SeqSetSetL := [T, {T=Sequence(S), S=Set(U,card >= 1), 
               U=Set(Z,card >= 1)},labeled]; A083355 := n-> count(SeqSetSetL,size=n);
%p A083355 A083355 := proc(n::integer) #with(combinat); local a,i,j; a:=0; for i 
               from 1 to n do for j from 1 to i do a := a + j!*stirling2(i,j)*stirling2(n,
               i); od; od; print("n, a(n): ",n, a); end proc; (Wieder)
%p A083355 A083355 := proc() local a,k,n; for n from 1 to 12 do a[n]:=0: for k from 
               1 to n do a[n]:=a[n]+stirling2(n,k)*A000670(k): od: od: print(a[1],
               a[2],a[3],a[4],a[5],a[6],a[7],a[8],a[9],a[10],a[11],a[12]); end proc; 
               A000670 := proc(n) local Result,k; Result:=0: for k from 1 to n do 
               Result:=Result+stirling2(n,k)*k! od: end proc;
%t A083355 Range[0, 18]!CoefficientList[Series[1/(2 - E^(E^x - 1)), {x, 0, 18}], 
               x] (from Robert G. Wilson v Jul 13 2004)
%o A083355 (PARI) a(n)=if(n<0,0,n!*polcoeff(1/(2-exp(exp(x+x*O(x^n))-1)),n))
%Y A083355 Cf. A055887.
%Y A083355 Cf. A000079, A000670, A055887, A034691, A075729.
%Y A083355 Sequence in context: A084357 A075729 A127131 this_sequence A141763 A025550 
               A067545
%Y A083355 Adjacent sequences: A083352 A083353 A083354 this_sequence A083356 A083357 
               A083358
%K A083355 nonn,new
%O A083355 0,3
%A A083355 Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 11 2003, May 07 2008