OFFSET
0,2
COMMENTS
a(n) also counts triangulations of a convex (2n+3)-gon whose points are colored red and blue alternatingly, and that do not have monochromatic triangles (i.e., every triangle has at least one red point and at least one blue point). - Torsten Muetze, May 08 2024
REFERENCES
Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, Ars Combinatoria, 88 (2008), 109-124.
LINKS
CombOS - Combinatorial Object Server, Generate k-ary trees and dissections
Bruce E. Sagan, Proper partitions of a polygon and k-Catalan numbers, arXiv:math/0407280 [math.CO], 2004.
FORMULA
a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(2*n+k+1,k) * binomial(5*n+3,n-2*k).
From Torsten Muetze, May 08 2024: (Start)
a(n) = 2^n/(n+1) * binomial(3n+1,n).
a(n) = 2^n*A006013(n). (End)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x*((1-x)^2-x^2)^2)/x)
(PARI) a(n) = sum(k=0, n\2, binomial(2*n+k+1, k)*binomial(5*n+3, n-2*k))/(n+1);
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Jan 25 2024
STATUS
approved