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 A102537 Triangle T(n,k) read by rows: 1/n * C(2n+k,k-1) * C(n,k). 4
 1, 1, 3, 1, 8, 12, 1, 15, 55, 55, 1, 24, 156, 364, 273, 1, 35, 350, 1400, 2380, 1428, 1, 48, 680, 4080, 11628, 15504, 7752, 1, 63, 1197, 9975, 41895, 92169, 100947, 43263, 1, 80, 1960, 21560, 123970, 396704, 708400, 657800, 246675, 1, 99, 3036, 42504 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Number of dissections of a convex (2n+2)-gon by k-1 noncrossing diagonals into (2j+2)-gons, 1 <= j <= n-1. Apparently, a signed, refined version of this array is given on page 65 of the Einziger link, related to the antipode of a Hopf algebra. - Tom Copeland, May 19 2015 The f-vectors of the simplicial noncrossing hypertree complexes of McCammond (p. 15). The reduced Euler characteristics are the signed Catalan numbers A000108. - Tom Copeland, May 19 2017 LINKS Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150). H. Einziger, Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions, Dissertation (2010), George Washington University. J. McCammond, Noncrossing Hypertrees, 2015. J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. E. Tzanaki, Polygon dissections and some generalizations of cluster complexes, arXiv:math/0501100 [math.CO], 2005. EXAMPLE Triangle begins   1;   1,  3;   1,  8,   12;   1, 15,   55,    55;   1, 24,  156,   364,    273;   1, 35,  350,  1400,   2380,   1428;   1, 48,  680,  4080,  11628,  15504,   7752;   1, 63, 1197,  9975,  41895,  92169, 100947,  43263;   1, 80, 1960, 21560, 123970, 396704, 708400, 657800, 246675; MATHEMATICA Table[1/n*Binomial[2 n + k, k - 1] Binomial[n, k], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, May 20 2017 *) PROG (MAGMA) [[1/n * Binomial(2*n+k, k-1) * Binomial(n, k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, May 20 2015 CROSSREFS Left-hand columns include A005563. Right-hand columns include essentially A001764 and A013698. Row sums are in A003168. Cf. A000108. Sequence in context: A120236 A049760 A019146 * A131202 A287987 A067955 Adjacent sequences:  A102534 A102535 A102536 * A102538 A102539 A102540 KEYWORD nonn,tabl AUTHOR Ralf Stephan, Jan 14 2005 STATUS approved

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Last modified August 18 11:51 EDT 2018. Contains 313832 sequences. (Running on oeis4.)