A001763 a(n) = (3n+3)!/(2n+3)!. (Formerly M4279 N1788)

1, 1, 6, 72, 1320, 32760, 1028160, 39070080, 1744364160, 89513424000, 5191778592000, 335885501952000, 23982224839372800, 1873278229119897600, 158905670470170624000, 14547557832075620352000, 1429628183315795054592000, 150110959248158480732160000

Offset

-1,3

Comments

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

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

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = -1..100

Anwar Al Ghabra, K. Gopala Krishna, Patrick Labelle, and Vasilisa Shramchenko, Enumeration of multi-rooted plane trees, arXiv:2301.09765 [math.CO], 2023.

L. W. Beineke and R. E. Pippert, Enumerating labeled k-dimensional trees and ball dissections, 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.

Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 407

Formula

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

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

Maple

A001763:=n->(3*n+3)!/(2*n+3)!: seq(A001763(n), n=-1..20); # Wesley Ivan Hurt, Aug 23 2014

Mathematica

Table[(3*n + 3)!/(2*n + 3)!, {n, -1, 20}] (* T. D. Noe, Aug 10 2012 *)

Crossrefs

Cf. A001762.

Sequence in context: A047058 A202382 A266869 * A003235 A113133 A302355

Adjacent sequences: A001760 A001761 A001762 * A001764 A001765 A001766

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Removed misleading phrase from definition as suggested by Allan Bickle. - N. J. A. Sloane, Feb 25 2024

Status

approved