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%I #78 Jan 20 2023 17:43:58
%S 1,4,25,216,2401,32768,531441,10000000,214358881,5159780352,
%T 137858491849,4049565169664,129746337890625,4503599627370496,
%U 168377826559400929,6746640616477458432,288441413567621167681,13107200000000000000000,630880792396715529789561
%N a(n) = (n+3)^n.
%C 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]
%C 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
%C 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
%C 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
%D 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.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.27.
%H T. D. Noe, <a href="/A007830/b007830.txt">Table of n, a(n) for n = 0..100</a>
%H Christian Brouder, William J. Keith, and Ângela Mestre, <a href="http://arxiv.org/abs/1301.0874">Closed forms for a multigraph enumeration</a>, arXiv preprint arXiv:1301.0874 [math.CO], 2013-2015.
%H P. J. Cameron, <a href="http://dx.doi.org/10.1016/0012-365X(92)00468-7">Two-graphs and Trees</a>, Discrete Math. 127 (1994) 63-74.
%H P. J. Cameron, <a href="https://doi.org/10.37236/1198">Counting two-graphs related to trees</a>, Elec. J. Combin., Vol. 2, #R4.
%H P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/groups.html">Sequences realized by oligomorphic permutation groups</a>, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
%H Vsevolod Gubarev, <a href="https://arxiv.org/abs/1811.08219">Rota-Baxter operators on a sum of fields</a>, arXiv:1811.08219 [math.RA], 2018.
%H Oleg Pikhurko, <a href="http://www.jstor.org/stable/30037632">Generating Edge-Labeled Trees</a>, American Math. Monthly, 112 (2005) 919-921.
%H M. Shapiro, B. Shapiro and A. Vainshtein, <a href="https://www.researchgate.net/publication/2656484_Ramified_Coverings_Of_S2_With_One_Degenerate_Branching_Point_And_Enumeration_Of_Edge-Ordered_Graphs/link/0c9605269e1eb83aae000000/download">Ramified coverings of S^2 with one degenerate branching point and enumeration of edge-ordered graphs</a>, 1996.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F 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
%F E.g.f.: -LambertW(-x)^3/(x^3 * (1+LambertW(-x))). - _Vladeta Jovovic_, Nov 07 2003
%F 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
%F E.g.f.: (1/2)*d/dx (LambertW(-x)/(-x))^2. - _Wolfdieter Lang_, Oct 25 2022
%p A007830:=n->(n+3)^n; seq(A007830(n), n=0..20);
%p T := -LambertW(-x): ser := series(exp(3*T)/(1-T), x, 20):
%p seq(n!*coeff(ser, x, n), n = 0..18); # _Peter Luschny_, Jan 20 2023
%t Table[(n+3)^n, {n, 0, 18}]
%o (PARI) a(n)=(n+3)^n \\ _Charles R Greathouse IV_, Feb 06 2017
%o (Magma) [(n+3)^n: n in [0..20]]; // _G. C. Greubel_, Mar 06 2020
%o (Sage) [(n+3)^n for n in (0..20)] # _G. C. Greubel_, Mar 06 2020
%Y Cf. A000169, A000272, A000312, A007778.
%Y Cf. A008785, A008786, A008787, A008788, A008789, A008790, A008791.
%K nonn,nice,easy
%O 0,2
%A _Peter J. Cameron_, Mar 15 1996
%E More terms from _Wesley Ivan Hurt_, May 05 2014
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