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A181517
Number of torsion pairs in the cluster category of type A_n.
2
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
3,2
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
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
Cf. A181519.
Sequence in context: A121545 A078845 A230126 * A110771 A082028 A373931
KEYWORD
easy,nonn
AUTHOR
Martin Rubey, Oct 26 2010
STATUS
approved