A000307 - OEIS

A000307

Number of 4-level labeled rooted trees with n leaves.
(Formerly M3590 N1455)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

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

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

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*)

CROSSREFS

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

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

Column k=3 of A144150.

Sequence in context: A152404 A062817 A196275 * A294346 A049376 A083410

Adjacent sequences: A000304 A000305 A000306 * A000308 A000309 A000310

KEYWORD

nonn,easy

AUTHOR