A110107 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n, having k return steps to the line y = x from the line y = x+1 or from the line y = x-1 (i.e., E steps from the line y = x+1 to the line y = x or N steps from the line y = x-1 to the line y = x).

1, 1, 2, 1, 8, 4, 1, 26, 28, 8, 1, 88, 136, 80, 16, 1, 330, 600, 512, 208, 32, 1, 1360, 2636, 2768, 1648, 512, 64, 1, 6002, 11892, 14024, 10544, 4832, 1216, 128, 1, 27760, 55376, 69728, 60768, 35712, 13312, 2816, 256, 1, 132690, 265200, 347072, 332768, 231232

Offset: 0

Author

Emeric Deutsch, Jul 11 2005

Keywords

• nonn
• tabl

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 sums are the central Delannoy numbers (A001850).

Examples

			T(2,1) = 8 because we have DN(E), DE(N), N(E)D, ND(E), NNE(E), E(N)D, ED(N) and EEN(N) (the return E or N steps are shown between parentheses).
Triangle begins:
  1;
  1,   2;
  1,   8,   4;
  1,  26,  28,   8;
  1,  88, 136,  80,  16;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• R. A. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.

Crossrefs

Cf. A001850, A110098, A110099.

Programs

Maple

R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/(1-z-2*t*z*R): Gser:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 9 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

Formula

Sum_{k=0..n} k*T(n,k) = 2*A110099(n).

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