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A007830 (n+3)^n. 19
1, 4, 25, 216, 2401, 32768, 531441, 10000000, 214358881, 5159780352, 137858491849, 4049565169664, 129746337890625, 4503599627370496, 168377826559400929, 6746640616477458432, 288441413567621167681, 13107200000000000000000, 630880792396715529789561 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(n-2) = 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 edge-labeled trees with n >= 2 edges is (n+1)^(n-2) by showing that the number of vertex-labeled trees of size n is n+1 times larger than the number of edge-labeled 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 edge-ordered graphs, Amer. Math. Soc. Transl., Vol. 180 (1997), pp. 219-227.

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, Two-graphs and Trees, Discrete Math. 127 (1994) 63-74.

P. J. Cameron, Counting two-graphs 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 Edge-Labeled Trees, American Math. Monthly, 112 (2005) 919-921.

Index entries for sequences related to trees

FORMULA

E.g.f. for b(n) = a(n-3): 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, A008785-A008791.

Sequence in context: A215094 A047733 A198198 * A218826 A060911 A060912

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(n-2) 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

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Last modified February 20 14:53 EST 2018. Contains 299380 sequences. (Running on oeis4.)