A000357 Number of 5-level labeled rooted trees with n leaves.

1, 1, 5, 35, 315, 3455, 44590, 660665, 11035095, 204904830, 4183174520, 93055783320, 2238954627848, 57903797748386, 1601122732128779, 47120734323344439, 1470076408565099152, 48449426629560437576, 1681560512531504058350, 61293054886119796799892

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.
• T. Hogg and B. A. Huberman, Attractors on finite sets: the dissipative dynamics of computing structures, Phys. Review A 32 (1985), 2338-2346.
• 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).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• 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. (Annotated scanned copy)
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 294
• Index entries for sequences related to rooted trees

Crossrefs

a(n)=|A039813(n,1)| (first column of triangle).

Cf. A000110, A000258, A000307, A000405, A001669.

Column k=4 of A144150.

Programs

Maple

g:= proc(p) local b; b:=proc(n) option remember; if n=0 then 1 else (n-1)! *add(p(k)*b(n-k)/ (k-1)!/ (n-k)!, k=1..n) fi end end: a:= g(g(g(g(1)))): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 11 2008

Mathematica

max = 17; Join[{1}, MatrixPower[Array[StirlingS2, {max, max}], 5][[All, 1]]] (* Jean-François Alcover, Mar 03 2014 *)

Formula

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

Extensions

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