A114687 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 double rises (0 <= k <= n-1).

1, 1, 2, 1, 6, 4, 1, 12, 24, 8, 1, 20, 80, 80, 16, 1, 30, 200, 400, 240, 32, 1, 42, 420, 1400, 1680, 672, 64, 1, 56, 784, 3920, 7840, 6272, 1792, 128, 1, 72, 1344, 9408, 28224, 37632, 21504, 4608, 256, 1, 90, 2160, 20160, 84672, 169344, 161280, 69120, 11520

Offset

1,3

Comments

Row sums are the little Schroeder numbers (A001003). Sum(k*T(n,k),k=0..n-1) = 2*A050152(n-1). Mirror image of A114656.

Triangle T(n,k) given (essentially) by [1,0,1,0,1,0,1,0,1,0,1,0,...] DELTA [0,2,0,2,0,2,0,2,0,2,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 02 2009

T(r, m) is the number distinct extremities of the [0,r]-covering hierarchies with segments terminating at r (see Kreweras work). - Michel Marcus, Nov 22 2014

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1 <= n <= 150).

D. Callan, Polygon Dissections and Marked Dyck Paths

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973), p. 23-24.

G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)

Formula

T(n, k) = 2^k * binomial(n, k) * binomial(n, k+1)/n.

G.f.: G=G(t, z) satisfies G = z*(1+G)*(1+2*t*G).

Example

T(3,2)=4 because we have UbUbUDDD, UbUrUDDD, UrUbUDDD and UrUrUDDD, where U=(1,1), D=(1,-1) and b (r) indicates a blue (red) double rise.
Triangle begins:
  1;
  1,  2;
  1,  6,  4;
  1, 12, 24,  8;
  1, 20, 80, 80, 16.
Triangle [1,0,1,0,1,0,1,0,...] DELTA [0,2,0,2,0,2,0,2,0,...]:= T(n,k), 0 <= k <= n, begins: 1; 1,0; 1,2,0; 1,6,4,0; 1,12,24,8,0; 1,20,80,80,16,0; ... - Philippe Deléham, Jan 02 2009

Maple

T:=(n, k)->2^k*binomial(n, k)*binomial(n, k+1)/n: for n from 1 to 11 do seq(T(n, k), k=0..n-1) od;

Mathematica

Table[2^k*Binomial[n, k] Binomial[n, k + 1]/n, {n, 10}, {k, 0, n - 1}] // Flatten (* Michael De Vlieger, Nov 05 2017 *)

Prog

(PARI) t(r, m) = 2^m*binomial(r, m)*binomial(r, m+1)/r;
tabl(nn) = {for (n=1, nn, for (k=0, n-1, print1(t(n, k), ", "); ); print(); ); } \\ Michel Marcus, Nov 22 2014

Crossrefs

Cf. A001003, A050152, A114656.

Sequence in context: A208761 A123519 A167024 * A137594 A112360 A250485

Adjacent sequences: A114684 A114685 A114686 * A114688 A114689 A114690

Keyword

nonn,tabl

Author

Emeric Deutsch, Dec 23 2005

Status

approved