A007830 a(n) = (n+3)^n.

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

Offset: 0

Author

Peter J. Cameron, Mar 15 1996

Keywords

• nonn
• nice
• easy

Comments

a(n-2) is the number of trees with n+1 unlabeled vertices and n labeled edges for n > 1. - Christian G. Bower, 12/99 [corrected by Jonathan Vos Post, Sep 22 2012]

a(n) is 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. - 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, and Ângela Mestre, Closed forms for a multigraph enumeration, arXiv preprint arXiv:1301.0874 [math.CO], 2013-2015.
• 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.
• Vsevolod Gubarev, Rota-Baxter operators on a sum of fields, arXiv:1811.08219 [math.RA], 2018.
• Oleg Pikhurko, Generating Edge-Labeled Trees, American Math. Monthly, 112 (2005) 919-921.
• M. Shapiro, B. Shapiro and A. Vainshtein, Ramified coverings of S^2 with one degenerate branching point and enumeration of edge-ordered graphs, 1996.
• Index entries for sequences related to trees

Crossrefs

Cf. A000169, A000272, A000312, A007778.

Cf. A008785, A008786, A008787, A008788, A008789, A008790, A008791.

Programs

Magma

[(n+3)^n: n in [0..20]]; // G. C. Greubel, Mar 06 2020

Maple

A007830:=n->(n+3)^n; seq(A007830(n), n=0..20);
T := -LambertW(-x): ser := series(exp(3*T)/(1-T), x, 20):
seq(n!*coeff(ser, x, n), n = 0..18); # Peter Luschny, Jan 20 2023

Mathematica

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

PARI

a(n)=(n+3)^n \\ Charles R Greathouse IV, Feb 06 2017

Sage

[(n+3)^n for n in (0..20)] # G. C. Greubel, Mar 06 2020

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/(x^3 * (1+LambertW(-x))). - Vladeta Jovovic, Nov 07 2003

With offset 1: E.g.f.: exp(T(x))^2/2 where T(x) is the e.g.f. for A000169. - Geoffrey Critzer, May 10 2013

E.g.f.: (1/2)*d/dx (LambertW(-x)/(-x))^2. - Wolfdieter Lang, Oct 25 2022

Extensions

More terms from Wesley Ivan Hurt, May 05 2014