OFFSET
3,2
COMMENTS
a(n) is also the number of Ptolemy diagrams on n vertices with distinguished base edge.
a(n) is the sum over all polygon dissections in a polygon with distinguished base edge, where each region of size at least four has weight two.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 3..500
Thorsten Holm, Peter Jorgensen, Martin Rubey, Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A_n, arXiv:1010.1184v1 [math.RT], 2010
FORMULA
G.f.: y*P(y) - y^2 where P(y) satisfies P(y) = y + P(y)^2*(1+P(y))/(1-P(y)).
EXAMPLE
For n=4 there are 4 Ptolemy diagrams: the square with no diagonal, two diagrams with one diagonal, and the square with both diagonals .
PROG
(PARI) a(n) = n-=3; sum(i=0, floor((n+1)/2), 2^i*binomial(n+1+i, i)*binomial(2*n+2, n+1-2*i))/(n+2); \\ Michel Marcus, Jan 14 2012; corrected Jun 13 2022
(PARI) seq(n) = Vec(x*(serreverse(x - x^2*(1 + x)/(1 - x) + O(x^(n+2))) - x)) \\ Andrew Howroyd, May 09 2023
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Martin Rubey, Oct 26 2010
STATUS
approved