A110123 Triangle read by rows: T(n,k) is the number of Delannoy paths of length n, having k EE's and NN's crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1 or two consecutive N steps from the line y = x-1 to the line y = x+1).

1, 3, 11, 2, 45, 16, 2, 197, 100, 22, 2, 903, 576, 174, 28, 2, 4279, 3206, 1202, 266, 34, 2, 20793, 17568, 7732, 2128, 376, 40, 2, 103049, 95592, 47676, 15452, 3408, 504, 46, 2, 518859, 518720, 286156, 105528, 27500, 5096, 650, 52, 2, 2646723, 2813514

Offset: 0

Author

Emeric Deutsch, Jul 13 2005

Keywords

• nonn
• tabf

Comments

A Delannoy path of length n is a path from (0,0) to (n,n), consisting of steps E=(1,0), N=(0,1) and D=(1,1).

Row 0 has one term; row n has n terms (n > 0).

Row sums are the central Delannoy numbers (A001850).

Column 0 yields the little Schroeder numbers (A001003).

Examples

			T(2,1)=2 because we have NEEN and ENNE.
Triangle begins:
    1;
    3;
   11,   2;
   45,  16,   2;
  197, 100,  22,   2;
		

Links

• Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Crossrefs

Cf. A001003, A001850, A006318, A100127, A110121.

Programs

Maple

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=simplify((1-z*R*t+z*R)/(1-z-z*R*t+z^2*R*t-z*R-z^2*R)): Gser:=simplify(series(G,z=0,14)): P[0]:=1: for n from 1 to 10 do P[n]:=coeff(Gser,z^n) od: 1; for n from 1 to 10 do seq(coeff(t*P[n],t^k),k=1..n) od; # yields sequence in triangular form

Formula

Sum_{k=0..n-1} k*T(n,k) = 2*A110127(n).

G.f.: (1 - tzR + zR)/(1 - z - tzR + tz^2*R - zR - z^2*R), where R = 1 + zR + zR^2 = (1 - z - sqrt(1 - 6z + z^2))/(2z) is the g.f. of the large Schroeder numbers (A006318).