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 A108410 Triangle T(n,k) read by rows: number of 12312-avoiding matchings on [2n] with exactly k crossings (n >= 1, 0 <= k <= n-1). 3
 1, 2, 1, 5, 5, 2, 14, 21, 15, 5, 42, 84, 84, 49, 14, 132, 330, 420, 336, 168, 42, 429, 1287, 1980, 1980, 1350, 594, 132, 1430, 5005, 9009, 10725, 9075, 5445, 2145, 429, 4862, 19448, 40040, 55055, 55055, 40898, 22022, 7865, 1430, 16796, 75582 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, arXiv:math/0504342 [math.CO], 2005. W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 2.2. D. S. Hough, Descents in noncrossing trees, Electronic J. Combinatorics 10 (2003), #N13, Theorem 2.2 [Ira M. Gessel, May 10 2010] FORMULA T(n, k) = Sum_{i=n..2n-1} (-1)^(n+k+i)/i*C(i, n)*C(3n, i+1+n)*C(i-n, k). T(n,k) = C(n-1+k,n-1)*C(2n-k,n+1)/n (0 <= k <= n-1). [Chen et al.] - Emeric Deutsch, Dec 19 2006 O.g.f. equals the series reversion w.r.t. x of x*(1 + x*(1 - t))/(1 + x)^3. If R(n,t) is the n-th row polynomial of this triangle then R(n 1+t) is the n-th row polynomial of A089434. - Peter Bala, Jul 15 2012 EXAMPLE Triangle begins      1;      2,     1;      5,     5,     2;     14,    21,    15,     5;     42,    84,    84,    49,    14;    132,   330,   420,   336,   168,    42;    429,  1287,  1980,  1980,  1350,   594,   132;   1430,  5005,  9009, 10725,  9075,  5445,  2145,  429;   4862, 19448, 40040, 55055, 55055, 40898, 22022, 7865, 1430; MAPLE T:=(n, k)->binomial(n-1+k, n-1)*binomial(2*n-k, n+1)/n: for n from 1 to 10 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form - Emeric Deutsch, Dec 19 2006 MATHEMATICA T[n_, k_] := Binomial[n + k - 1, n - 1]*Binomial[2*n - k, n + 1]/n; Table[T[n, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Nov 11 2017, after Emeric Deutsch *) PROG (PARI) T(n, k) = binomial(n-1+k, n-1)*binomial(2*n-k, n+1)/n; \\ Andrew Howroyd, Nov 06 2017 CROSSREFS Left-hand columns include A000108 and A002054. Right-hand columns include A000108 and A007851+1. Row sums are A001764. A089434. Sequence in context: A046757 A248905 A118244 * A058116 A058118 A124226 Adjacent sequences:  A108407 A108408 A108409 * A108411 A108412 A108413 KEYWORD nonn,tabl AUTHOR Ralf Stephan, Jun 03 2005 STATUS approved

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Last modified January 17 18:48 EST 2019. Contains 319251 sequences. (Running on oeis4.)