A118964 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that have k double rises above the x-axis (n >= 1, k >= 0).

2, 5, 1, 14, 5, 1, 42, 19, 8, 1, 132, 67, 40, 12, 1, 429, 232, 166, 79, 17, 1, 1430, 804, 634, 395, 145, 23, 1, 4862, 2806, 2335, 1708, 879, 249, 30, 1, 16796, 9878, 8480, 6824, 4376, 1823, 404, 38, 1, 58786, 35072, 30691, 26137, 19334, 10521, 3542, 625, 47, 1

Offset: 1

Author

Emeric Deutsch, May 07 2006

Keywords

• nonn
• tabl

Comments

A Grand Dyck path of semilength n is a path in the half-plane x >= 0, starting at (0,0), ending at (2n,0) and consisting of steps u = (1,1) and d = (1,-1); a double rise in a Grand Dyck path is an occurrence of uu in the path.

For all double rises (above, below and on the x-axis), see A118963.

Examples

			T(3,1) = 5 because we have u/ududd,u/uddud,udu/udd,duu/udd and u/udddu (the double rises above the x-axis are indicated by /).
Triangle starts:
  2;
  5,   1;
  14,  5,  1;
  42,  19, 8,  1;
  132, 67, 40, 12, 1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

T(n,0) = A000108(n+1) (Catalan numbers), T(n,1) = A114277(n-2).

Cf. A000984 (row sums), A000108, A000531, A118963.

Programs

Maple

C:=(1-sqrt(1-4*z))/2/z: r:=(1-z-t*z-sqrt(z^2*t^2-2*z^2*t-2*z*t+z^2-2*z+1))/2/t/z: G:=(1+r)/(1-z*C*(1+r))-1: Gser:=simplify(series(G,z=0,15)): for n from 1 to 11 do P[n]:=coeff(Gser,z,n) od: for n from 1 to 11 do seq(coeff(P[n],t,j),j=0..n-1) od; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(abs(y)>x, 0,
      `if`(x=0, 1, expand(`if`(t=2, z, 1)*b(x-1, y+1,
      `if`(y>=0, min(t+1, 2), 1)) +b(x-1, y-1, 1))))
    end:
T:= n-> (p-> seq(coeff(p,z,i), i=0..n-1))(b(2*n, 0, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Jun 16 2014

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[Abs[y] > x, 0, If[x == 0, 1, Expand[If[t == 2, z, 1]*b[x-1, y+1, If[y >= 0, Min[t+1, 2], 1]] + b[x-1, y-1, 1]]]]; T[n_] := Function[ {p}, Table[Coefficient[p, z, i], {i, 0, n-1}]][b[2*n, 0, 1]]; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

Formula

Sum_{k>=0} k*T(n,k) = A000531(n-1).

G.f.: G(t,z) = (1+r)/[1-z(1+r)C]-1, where r = r(t,z) is the Narayana function, defined by (1+r)(1+tr)z = r, r(t,0) = 0 and C = C(z) = [1-sqrt(1-4z)]/(2z) is the Catalan function. More generally, the g.f. H = H(t,s,u,z), where t,s and u mark double rises above, below and on the x-axis, respectively, is H = [1 + r(s,z)]/[1 - z(1 + tr(t,z))(1 + ur(s,z))].

Extensions

Keyword tabf changed to tabl by Michel Marcus, Apr 07 2013