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%I M2932 N1178 #121 Oct 27 2023 18:22:23
%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 Expansion of 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 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, and 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 Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, <a href="https://arxiv.org/abs/2107.04170">Brauer and Jones tied monoids</a>, arXiv:2107.04170 [math.RT], 2021.
%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 A. Joseph Kennedy, P. Jaish, and P. Sundaresan, <a href="https://doi.org/10.26637/MJM0802/0009">Note on generating function of higher dimensional bell numbers</a> (sic), Malaya Journal of Matematik (2020) Vol.8, No. 2, 369-372.
%H Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H T. Mansour, A. Munagi, and 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="https://arxiv.org/abs/math/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_{k=0..n} Stirling2(n, k)*Bell(k). - 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
%F G.f.: Sum_{j>=0} Bell(j)*x^j / Product_{k=1..j} (1 - k*x). - _Ilya Gutkovskiy_, Apr 06 2019
%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];
%p # alternative Maple program:
%p b:= proc(n, t) option remember; `if`(n=0 or t=0, 1, add(
%p b(n-j, t)*b(j, t-1)*binomial(n-1, j-1), j=1..n))
%p end:
%p a:= n-> b(n, 2):
%p seq(a(n), n=0..23); # _Alois P. Heinz_, Sep 02 2021
%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-François 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 */
%o (Magma) m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(Exp(Exp(x)-1)-1))); [Factorial(n-1)*b[n]: n in [1..m]]; // _Vincenzo Librandi_, Feb 01 2020
%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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