

A001763


Number of dissections of a ball: (3n+3)!/(2n+3)!.
(Formerly M4279 N1788)


5



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


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
L. W. Beineke and R. E. Pippert, Enumerating labeled kdimensional trees and ball dissections, pp. 1226 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), 8798.
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)*x2*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.


STATUS

approved



