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 A114656 Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n). 4
 1, 2, 1, 4, 6, 1, 8, 24, 12, 1, 16, 80, 80, 20, 1, 32, 240, 400, 200, 30, 1, 64, 672, 1680, 1400, 420, 42, 1, 128, 1792, 6272, 7840, 3920, 784, 56, 1, 256, 4608, 21504, 37632, 28224, 9408, 1344, 72, 1, 512, 11520, 69120, 161280, 169344, 84672, 20160, 2160, 90 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums are the little Schroeder numbers (A001003). Sum_{k=1..n} k*T(n,k) = A047781(n). T(n,k) = (1/2)A114655(n,k). Triangle T(n,k), 1 <= k <= n, given by [0,2,0,2,0,2,0,2,0,2,0,2,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 02 2009 LINKS Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150, flattened). D. Callan, Polygon Dissections and Marked Dyck Paths Samuele Giraudo, Tree series and pattern avoidance in syntax trees, arXiv:1903.00677 [math.CO], 2019. FORMULA T(n, k) = 2^(n-k)*binomial(n, k)*binomial(n, k-1)/n. G.f.: G = G(t, z) satisfies G = z(2G+t)(G+1). T(n,k) = A001263(n,k)*2^(n-k). - Philippe Deléham, Apr 11 2007 G.f.: 1/(1-xy/(1-2x/(1-xy/(1-2x/(1-xy/(1-2x/(1-..... (continued fraction). - Paul Barry, Feb 06 2009 EXAMPLE T(3,2)=6 because we have (UD)Ub(UD)D, (UD)Ur(UD)D, Ub(UD)D(UD), Ur(UD)D(UD), Ub(UD)(UD)D and Ur(UD)(UD)D, where U=(1,1), D=(1,-1) and b (r) indicates a blue (red) double rise (the peaks are shown between parentheses). Triangle begins: 1; 2, 1; 4, 6, 1; 8, 24, 12, 1; 16, 80, 80, 20, 1; .... Triangle T(n,k), 0 <= k <= n, given by [0,2,0,2,0,2,0,2,...] DELTA [1,0,1,0,1,0,1,0,1,0,...] begins: 1; 0,1; 0,2,1; 0,4,6,1; 0,8,24,12,1; 0,16,80,80,20,1; ... - Philippe Deléham, Jan 02 2009 MAPLE T:=(n, k)->2^(n-k)*binomial(n, k)*binomial(n, k-1)/n: for n from 1 to 11 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form MATHEMATICA Table[2^(n - k) Binomial[n, k] Binomial[n, k - 1]/n, {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Apr 23 2019 *) CROSSREFS Cf. A001003, A047781, A114655. Sequence in context: A274292 A359670 A114192 * A294440 A346905 A075497 Adjacent sequences: A114653 A114654 A114655 * A114657 A114658 A114659 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, Dec 23 2005 STATUS approved

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Last modified October 3 10:55 EDT 2023. Contains 365861 sequences. (Running on oeis4.)