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%I #21 May 09 2023 18:26:42
%S 1,4,17,82,422,2274,12665,72326,421214,2492112,14937210,90508256,
%T 553492552,3411758334,21175624713,132226234854,830077057878,
%U 5235817447752,33166634502334,210904780742860,1345806528336772,8614979593487972,55307373497626442,356012579697723084
%N Number of torsion pairs in the cluster category of type A_n.
%C a(n) is also the number of Ptolemy diagrams on n vertices with distinguished base edge.
%C 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.
%H Andrew Howroyd, <a href="/A181517/b181517.txt">Table of n, a(n) for n = 3..500</a>
%H Thorsten Holm, Peter Jorgensen, Martin Rubey, <a href="http://arxiv.org/abs/1010.1184">Ptolemy diagrams and torsion pairs in the cluster category of Dynkin type A_n</a>, arXiv:1010.1184v1 [math.RT], 2010
%F G.f.: y*P(y) - y^2 where P(y) satisfies P(y) = y + P(y)^2*(1+P(y))/(1-P(y)).
%e For n=4 there are 4 Ptolemy diagrams: the square with no diagonal, two diagrams with one diagonal, and the square with both diagonals .
%o (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
%o (PARI) seq(n) = Vec(x*(serreverse(x - x^2*(1 + x)/(1 - x) + O(x^(n+2))) - x)) \\ _Andrew Howroyd_, May 09 2023
%Y Cf. A181519.
%K easy,nonn
%O 3,2
%A _Martin Rubey_, Oct 26 2010