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

1,2

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