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 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 (list; graph; refs; listen; history; text; internal format)
 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 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. 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 STATUS approved

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Last modified September 24 07:30 EDT 2018. Contains 315308 sequences. (Running on oeis4.)