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%I M3590 N1455 #61 Aug 06 2018 14:54:43
%S 1,1,4,22,154,1304,12915,146115,1855570,26097835,402215465,6734414075,
%T 121629173423,2355470737637,48664218965021,1067895971109199,
%U 24795678053493443,607144847919796830,15630954703539323090,421990078975569031642,11918095123121138408128
%N Number of 4-level labeled rooted trees with n leaves.
%D J. de la Cal, J. Carcamo, Set partitions and moments of random variables, J. Math. Anal. Applic. 378 (2011) 16 doi:10.1016/j.jmaa.2011.01.002 Remark 5
%D J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
%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="/A000307/b000307.txt">Table of n, a(n) for n = 0..440</a>
%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 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=293">Encyclopedia of Combinatorial Structures 293</a>
%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.
%H K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="https://doi.org/10.1088/0305-4470/37/10/011">Hierarchical Dobinski-type relations via substitution and the moment problem</a>, J. Phys. A: Math.Gen 37 (2004) 3475-3487.
%H John Riordan, <a href="/A002720/a002720_2.pdf">Letter, Apr 28 1976.</a>
%H Kruchinin Vladimir Victorovich, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F E.g.f.: exp(exp(exp(exp(x)-1)-1)-1).
%F a(n) = sum(sum(sum(stirling2(n,k) *stirling2(k,m) *stirling2(m,r), k=m..n), m=r..n), r=1..n), n>0. - _Vladimir Kruchinin_, Sep 08 2010
%p g:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, (n-1)! *add(p(k)*b(n-k)/ (k-1)!/ (n-k)!, k=1..n)) end end: a:= g(g(g(1))): seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 11 2008
%t nn = 18; a = Exp[Exp[x] - 1]; b = Exp[a - 1];
%t Range[0, nn]! CoefficientList[Series[Exp[b - 1], {x, 0, nn}], x] (*_Geoffrey Critzer_, Dec 28 2011*)
%Y a(n)=|A039812(n,1)| (first column of triangle).
%Y Cf. A000110, A000258, A000357, A000405, A001669.
%Y Column k=3 of A144150.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E Extended with new definition by _Christian G. Bower_, Aug 15 1998
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