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
MATHEMATICA
Table[(3*n + 3)!/(2*n + 3)!, {n, -1, 20}] (* T. D. Noe, Aug 10 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Removed misleading phrase from definition as suggested by Allan Bickle. - N. J. A. Sloane, Feb 25 2024
STATUS
approved