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%I M4279 N1788 #47 Feb 25 2024 11:07:52
%S 1,1,6,72,1320,32760,1028160,39070080,1744364160,89513424000,
%T 5191778592000,335885501952000,23982224839372800,1873278229119897600,
%U 158905670470170624000,14547557832075620352000,1429628183315795054592000,150110959248158480732160000
%N a(n) = (3n+3)!/(2n+3)!.
%C With offset 1, a(n) = number of labeled plane trees (A006963) on n vertices in which vertices of degree d come in d colors or, equivalently, each vertex has a favorite neighbor (n>=2). For example, there are 2 unlabeled plane trees with 4 vertices: the path and the star. There are 4!/2 ways to label the path and 4!/3 ways to label the star. There are 4 choices for coloring vertices in the path and 3 choices for coloring vertices in the star. The count for 4 vertices is thus 12*4 + 8*3 = 72. - _David Callan_, Aug 22 2014
%C This is the number of labeled Apollonian networks (planar 3-trees) with n+4 vertices rooted at an exterior triangle. - _Allan Bickle_, Feb 20 2024
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001763/b001763.txt">Table of n, a(n) for n = -1..100</a>
%H Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, <a href="https://arxiv.org/abs/2301.09765">Enumeration of multi-rooted plane trees</a>, arXiv:2301.09765 [math.CO], 2023.
%H L. W. Beineke and R. E. Pippert, <a href="http://dx.doi.org/ 10.1007/BF02330563">Enumerating labeled k-dimensional trees and ball dissections</a>, pp. 12-26 of Proceedings of Second Chapel Hill Conference on Combinatorial Mathematics and Its Applications, University of North Carolina, Chapel Hill, 1970. Reprinted in Math. Annalen, 191 (1971), 87-98.
%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=407">Encyclopedia of Combinatorial Structures 407</a>
%F E.g.f.: A(x) = (2/sqrt(3*x))*sin(arcsin(3*sqrt(3*x)/2)/3) = 1+6*x/(Q(0)-6*x); Q(k) = 3*x*(3*k+1)*(3*k+2) + 2*(2*(k^2)+5*k+3) - 6*x*(2*(k^2)+5*k+3)*(3*k+4)*(3*k+5)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 27 2011
%F E.g.f. (starting at n=0 term): -(1/3)*(3*cos((2/3)*arcsin((3/2)*3^(1/2)*x^(1/2)))*x^(1/2)*(-27*x+4)^(1/2) + 9*sin((2/3)*arcsin((3/2)*3^(1/2)*x^(1/2)))*3^(1/2)*x - 2*sin((2/3)*arcsin((3/2)*3^(1/2)*x^(1/2)))*3^(1/2))/(x^(3/2)*(-27*x+4)^(1/2)). - _Robert Israel_, Aug 22 2014
%p A001763:=n->(3*n+3)!/(2*n+3)!: seq(A001763(n), n=-1..20); # _Wesley Ivan Hurt_, Aug 23 2014
%t Table[(3*n + 3)!/(2*n + 3)!, {n, -1, 20}] (* _T. D. Noe_, Aug 10 2012 *)
%Y Cf. A001762.
%K nonn
%O -1,3
%A _N. J. A. Sloane_
%E Removed misleading phrase from definition as suggested by _Allan Bickle_. - _N. J. A. Sloane_, Feb 25 2024
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