%I A000258 M2932 N1178
%S A000258 1,1,3,12,60,358,2471,19302,167894,1606137,16733779,188378402,
%T A000258 2276423485,29367807524,402577243425,5840190914957,89345001017415,
%U A000258 1436904211547895,24227076487779802,427187837301557598
%N A000258 E.g.f.: exp(exp(exp(x)-1)-1).
%C A000258 Number of 3-level labeled rooted trees with n leaves. - Christian G. 
               Bower (bowerc(AT)usa.net), Aug 15 1998
%C A000258 Number of pairs of set partitions (d,d') of [n] such that d is finer 
               than d'. - A. Joseph Kennedy (kennedy_2001in(AT)yahoo.co.in), Feb 
               05 2006
%C A000258 In the Comm. Algebra paper cited, I introduce a sequence of algebras 
               called 'class partition algebras' with this sequence as dimensions. 
               The algebras are the centralizers of wreath product in combinatorial 
               representation theory. - A. Joseph Kennedy (kennedy_2001in(AT)yahoo.co.in), 
               Feb 17 2008
%D A000258 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000258 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000258 J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
%D A000258 T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative 
               dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
%D A000258 A. Joseph Kennedy, Class partition algebras as centralizer algebras, 
               Communications in Algebra, 35 (2007), 145-170, see page 153.
%D A000258 R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see 
               Example 5.2.4.
%H A000258 T. D. Noe, <a href="b000258.txt">Table of n, a(n) for n=0..100</a>
%H A000258 P. Blasiak, A. Horzela, K. A. Penson, G.H.E. Duchamp and A. I. Solomon, 
               <a href="http://arXiv.org/abs/quant-ph/0501155">Boson normal ordering 
               via substitutions and Sheffer-type polynomials</a>
%H A000258 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000258 P. J. Cameron, D. A. Gewurz and F. Merola, <a href="http://www.maths.qmw.ac.uk/
               ~pjc/preprints/product.pdf">Product action</a>, Discrete Math., 308 
               (2008), 386-394.
%H A000258 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=70">
               Encyclopedia of Combinatorial Structures 70</a>
%H A000258 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=292">
               Encyclopedia of Combinatorial Structures 292</a>
%H A000258 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 A000258 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 A000258 <a href="Sindx_Ro.html#rooted">Index entries for sequences related to 
               rooted trees</a>
%H A000258 Gottfried Helms, <a href="http://go.helms-net.de/math/binomial/04_5_SummingBellStirling.pdf">
               Bell Numbers</a>, 2008.
%F A000258 Sum(stirling2(n, k)*(bell(k)), k=0..n). - Detlef Pauly (dettodet(AT)yahoo.de), 
               Jun 06 2002
%F A000258 Representation as an infinite series (Dobinski-type formula), in Maple 
               notation: a(n)=exp(exp(-1)-1)*sum(evalf(sum(p!*stirling2(k, p)*exp(-p), 
               p=1..k))*k^n/k!, k=0..infinity), n=1, 2.... - Karol A. Penson (penson(AT)lptl.jussieu.fr), 
               Nov 28 2003.
%F A000258 a(n)=sum_{k=0..n) A055896(n,k). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Apr 15 2008
%p A000258 with(combinat, bell, stirling2): seq(add(stirling2(n,k)*(bell(k)), k=0..n),
               n=0..30);
%p A000258 with(combstruct); SetSetSetL := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,
               card >=1)},labeled];
%t A000258 NestList[ Factor[ D[ #1, x ] ]&, Exp[ Exp[ Exp[ x-1 ]-1 ]-1 ], n ] /. 
               (x->1)
%Y A000258 a(n)=|A039811(n, 1)| (first column of triangle). Cf. A000110, A000307, 
               A000357, A000405, A001669.
%Y A000258 Row sums of (Stirling2)^2 triangle A130191.
%Y A000258 Sequence in context: A096471 A140097 A105227 this_sequence A070863 A062569 
               A089057
%Y A000258 Adjacent sequences: A000255 A000256 A000257 this_sequence A000259 A000260 
               A000261
%K A000258 nonn,easy,nice
%O A000258 0,3
%A A000258 N. J. A. Sloane (njas(AT)research.att.com).