A181517 Number of torsion pairs in the cluster category of type A_n.

1, 4, 17, 82, 422, 2274, 12665, 72326, 421214, 2492112, 14937210, 90508256, 553492552, 3411758334, 21175624713, 132226234854, 830077057878, 5235817447752, 33166634502334, 210904780742860, 1345806528336772, 8614979593487972, 55307373497626442, 356012579697723084

Offset: 3

Author

Martin Rubey, Oct 26 2010

Keywords

• easy
• nonn

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.

Examples

			For n=4 there are 4 Ptolemy diagrams: the square with no diagonal, two diagrams with one diagonal, and the square with both diagonals .
		

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

Crossrefs

Cf. A181519.

Programs

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

Formula

G.f.: y*P(y) - y^2 where P(y) satisfies P(y) = y + P(y)^2*(1+P(y))/(1-P(y)).