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 A181517 Number of torsion pairs in the cluster category of type A_n. 1
 1, 4, 17, 82, 422, 2274, 12665, 72326, 421214, 2492112, 14937210, 90508256, 553492552, 3411758334, 21175624713, 132226234854, 830077057878, 5235817447752, 33166634502334, 210904780742860, 1345806528336772 (list; graph; refs; listen; history; text; internal format)
 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. 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) = 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 CROSSREFS Cf. A181519. Sequence in context: A121545 A078845 A230126 * A110771 A082028 A052315 Adjacent sequences:  A181514 A181515 A181516 * A181518 A181519 A181520 KEYWORD easy,nonn AUTHOR Martin Rubey (martin.rubey(AT)math.uni-hannover.de), Oct 26 2010 STATUS approved

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Last modified January 26 16:30 EST 2021. Contains 340442 sequences. (Running on oeis4.)