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A181517
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Number of torsion pairs in the cluster category of type A_n.
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2
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1, 4, 17, 82, 422, 2274, 12665, 72326, 421214, 2492112, 14937210, 90508256, 553492552, 3411758334, 21175624713, 132226234854, 830077057878, 5235817447752, 33166634502334, 210904780742860, 1345806528336772, 8614979593487972, 55307373497626442, 356012579697723084
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OFFSET
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3,2
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f.: y*P(y) - y^2 where P(y) satisfies P(y) = y + P(y)^2*(1+P(y))/(1-P(y)).
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EXAMPLE
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For n=4 there are 4 Ptolemy diagrams: the square with no diagonal, two diagrams with one diagonal, and the square with both diagonals .
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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