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A000258 E.g.f.: exp(exp(exp(x)-1)-1).
(Formerly M2932 N1178)

%I M2932 N1178

%S 1,1,3,12,60,358,2471,19302,167894,1606137,16733779,188378402,

%T 2276423485,29367807524,402577243425,5840190914957,89345001017415,

%U 1436904211547895,24227076487779802,427187837301557598,7859930038606521508,150601795280158255827

%N E.g.f.: exp(exp(exp(x)-1)-1).

%C Number of 3-level labeled rooted trees with n leaves. - _Christian G. Bower_, Aug 15 1998

%C 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 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

%C a(n) is the number of ways to partition {1,2,...,n} and then partition each cell (block) into subcells.

%D J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.

%D Ulrike Sattler, Decidable classes of formal power series with nice closure properties, Diplomarbeit im Fach Informatik, Univ. Erlangen - Nuernberg, Jul 27 1994

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.4.

%H T. D. Noe and Alois P. Heinz, <a href="/A000258/b000258.txt">Table of n, a(n) for n = 0..475</a> (first 101 terms from T. D. Noe)

%H A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, O. Mallet, <a href="http://arxiv.org/abs/1402.2960">Bell polynomials in combinatorial Hopf algebras</a>, arXiv preprint arXiv:1402.2960 [math.CO], 2014.

%H 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>, arXiv:quant-ph/0501155, 2005.

%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

%H 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 Jekuthiel Ginsburg, <a href="/A000405/a000405.pdf">Iterated exponentials</a>, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]

%H Gottfried Helms, <a href="http://go.helms-net.de/math/binomial/04_5_SummingBellStirling.pdf">Bell Numbers</a>, 2008.

%H T. Hogg and B. A. Huberman, <a href="http://link.aps.org/doi/10.1103/PhysRevA.32.2338">Attractors on finite sets: the dissipative dynamics of computing structures</a>, Phys. Review A 32 (1985), 2338-2346.

%H T. Hogg and B. A. Huberman, <a href="/A000258/a000258.pdf">Attractors on finite sets: the dissipative dynamics of computing structures</a>, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=70">Encyclopedia of Combinatorial Structures 70</a>

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=292">Encyclopedia of Combinatorial Structures 292</a>

%H A. Joseph Kennedy, <a href="http://dx.doi.org/10.1080/00927870601041441">Class partition algebras as centralizer algebras</a>, Communications in Algebra, 35 (2007), 145-170, see page 153.

%H T. Mansour, A. Munagi, M. Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Mansour/mansour4.html">Recurrence Relations and Two-Dimensional Set Partitions </a>, J. Int. Seq. 14 (2011) # 11.4.1

%H 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>, arXiv:quant-ph/0312202, 2003, [J. Phys. A 37 (2004), 3475-3487].

%H John Riordan, <a href="/A002720/a002720_2.pdf">Letter, Apr 28 1976.</a>

%H N. J. A. Sloane and Thomas Wieder, <a href="http://arXiv.org/abs/math.CO/0307064">The Number of Hierarchical Orderings</a>, arXiv:math/0307064 [math.CO], 2003, [Order 21 (2004), 83-89].

%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>

%F a(n) = |A039811(n, 1)| (first column of triangle).

%F a(n) = Sum(stirling2(n, k)*(bell(k)), k=0..n). - Detlef Pauly (dettodet(AT)yahoo.de), Jun 06 2002

%F 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_, Nov 28 2003.

%F a(n) = Sum_{k=0..n} A055896(n,k). - _R. J. Mathar_, Apr 15 2008

%e G.f. = 1 + x + 3*x^2 + 12*x^3 + 60*x^4 + 358*x^5 + 2471*x^6 + 19302*x^7 + ...

%p with(combinat, bell, stirling2): seq(add(stirling2(n,k)*(bell(k)), k=0..n),n=0..30);

%p with(combstruct); SetSetSetL := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},labeled];

%t nn = 20; Range[0, nn]! CoefficientList[Series[Exp[Exp[Exp[x] - 1] - 1], {x, 0, nn}], x]

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ Exp[ Exp[x] - 1] - 1] , {x, 0, n}]]; (* _Michael Somos_, Aug 15 2015 *)

%t a[n_] := Sum[StirlingS2[n, k]*BellB[k], {k, 0, n}]; Table[a[n], {n, 0, 25}] (* _Jean-Fran├žois Alcover_, Feb 06 2016 *)

%t Table[Sum[BellY[n, k, BellB[Range[n]]], {k, 0, n}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)

%o (Maxima) makelist(sum(stirling2(n,k)*belln(k),k,0,n),n,0,24); /* _Emanuele Munarini_, Jul 04 2011 */

%Y Cf. A000110, A000307, A000357, A000405, A001669, A039811.

%Y Row sums of (Stirling2)^2 triangle A130191.

%Y Column k=2 of A144150.

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_

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Last modified February 23 04:30 EST 2019. Contains 320411 sequences. (Running on oeis4.)