A000307 Number of 4-level labeled rooted trees with n leaves.

1, 1, 4, 22, 154, 1304, 12915, 146115, 1855570, 26097835, 402215465, 6734414075, 121629173423, 2355470737637, 48664218965021, 1067895971109199, 24795678053493443, 607144847919796830, 15630954703539323090, 421990078975569031642, 11918095123121138408128

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• 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
• J. Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.4.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..440
• P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
• Jekuthiel Ginsburg, Iterated exponentials, Scripta Math., 11 (1945), 340-353. [Annotated scanned copy]
• Gottfried Helms, Bell Numbers, 2008.
• T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
• T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346. (Annotated scanned copy)
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 293
• K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, arXiv:quant-ph/0312202, 2003.
• K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem, J. Phys. A: Math.Gen 37 (2004) 3475-3487.
• John Riordan, Letter, Apr 28 1976.
• Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
• Index entries for sequences related to rooted trees

Crossrefs

a(n)=|A039812(n,1)| (first column of triangle).

Cf. A000110, A000258, A000357, A000405, A001669.

Column k=3 of A144150.

Programs

Maple

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

Mathematica

nn = 18; a = Exp[Exp[x] - 1]; b = Exp[a - 1];
Range[0, nn]! CoefficientList[Series[Exp[b - 1], {x, 0, nn}], x]  (*Geoffrey Critzer, Dec 28 2011*)

Formula

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

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

Extensions

Extended with new definition by Christian G. Bower, Aug 15 1998