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).
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
Offset: 1
Examples
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
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.
Programs
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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
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Mathematica
Table[2^(n - k) Binomial[n, k] Binomial[n, k - 1]/n, {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Apr 23 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
Comments