



1, 4, 25, 216, 2401, 32768, 531441, 10000000, 214358881, 5159780352, 137858491849, 4049565169664, 129746337890625, 4503599627370496, 168377826559400929, 6746640616477458432, 288441413567621167681, 13107200000000000000000, 630880792396715529789561
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OFFSET

0,2


COMMENTS

a(n2) = number of trees with n+1 unlabeled vertices and n labeled edges for n > 1.  Christian G. Bower, 12/99.
a(n) is also the number of nonequivalent primitive meromorphic functions with one pole of order n+3 on a Riemann surface of genus 0.  Noam Katz (noamkj(AT)hotmail.com), Mar 30 2001
Pikhurko writes: "Cameron demonstrated that the total number of edgelabeled trees with n >= 2 edges is (n+1)^(n2) by showing that the number of vertexlabeled trees of size n is n+1 times larger than the number of edgelabeled ones."  Jonathan Vos Post, Sep 22 2012
With offset 1, a(n) is the number of ways to build a rooted labeled forest with some (possibly all or none) of the nodes from {1,2,...,n} and then build another forest with the remaining nodes. E.g.f. is exp(T(x))^2/2 where T(x) is the e.g.f. for A000169.  Geoffrey Critzer, May 10 2013


REFERENCES

M. Shapiro, B. Shapiro and A. Vainshtein  Ramified coverings of S^2 with one degenerate branching point and enumeration of edgeordered graphs, Amer. Math. Soc. Transl., Vol. 180 (1997), pp. 219227.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.27.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..100
Christian Brouder, William J. Keith, Ângela Mestre, Closed forms for a multigraph enumeration, arXiv preprint arXiv:1301.0874, 2013.
P. J. Cameron, Twographs and Trees, Discrete Math. 127 (1994) 6374.
P. J. Cameron, Counting twographs related to trees, Elec. J. Combin., Vol. 2, #R4.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Oleg Pikhurko, Generating EdgeLabeled Trees, American Math. Monthly, 112 (2005) 919921.
Index entries for sequences related to trees


FORMULA

E.g.f. for b(n) = a(n3): T(x)(3/4)T^2(x)+(1/6)T^3(x), where T(x) is Euler's tree function (see A000169).  Len Smiley, Nov 17 2001
E.g.f.: LambertW(x)^3/(1+LambertW(x))/x^3.  Vladeta Jovovic, Nov 07 2003


MAPLE

A007830:=n>(n+3)^n; seq(A007830(n), n=0..20);


MATHEMATICA

Table[ (n+3)^n, {n, 0, 18} ]


PROG

(? Language ?) for n to 6 do ST := [seq(seq(i, j = 1 .. n+2), i = 1 .. n)]; PST := powerset(ST);
Result[n] := nops(PST) end do; seq(Result[n], n = 1 .. 6)  Thomas Wieder, Feb 07 2010
(PARI) a(n)=(n+3)^n \\ Charles R Greathouse IV, Feb 06 2017


CROSSREFS

Cf. A000169, A000272, A000312, A007778, A008785A008791.
Sequence in context: A215094 A047733 A198198 * A305404 A218826 A060911
Adjacent sequences: A007827 A007828 A007829 * A007831 A007832 A007833


KEYWORD

nonn,nice,easy


AUTHOR

Peter J. Cameron, Mar 15 1996


EXTENSIONS

Corrected the comment about number of trees with n+1 unlabeled vertices and n labeled edges. This number is a(n2) and not a(n+2) as originally stated.  Jonathan Vos Post, Sep 22 2012
More terms from Wesley Ivan Hurt, May 05 2014


STATUS

approved



