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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 Alexander Burstein, Megan Martinez, Pattern classes equinumerous to the class of ternary forests, Permutation Patterns Virtual Workshop, Howard University (2020). 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..2*n-1} (-1)^(n+k+i)/i*C(i, n)*C(3*n, i+1+n)*C(i-n, k). T(n,k) = C(n-1+k,n-1)*C(2*n-k,n+1)/n, (0 <= k <= n-1). [Chen et al.] - Emeric Deutsch, Dec 19 2006 O.g.f. equals the series reversion with respect to 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 A332632 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 November 30 05:46 EST 2021. Contains 349419 sequences. (Running on oeis4.)