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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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1
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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
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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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Table of n, a(n) for n=3..23.
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
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FORMULA
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G.F. 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) = 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
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CROSSREFS
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Cf. A181519.
Sequence in context: A121545 A078845 A230126 * A110771 A082028 A052315
Adjacent sequences: A181514 A181515 A181516 * A181518 A181519 A181520
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KEYWORD
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easy,nonn
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AUTHOR
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Martin Rubey (martin.rubey(AT)math.uni-hannover.de), Oct 26 2010
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STATUS
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approved
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