A110221 Triangle read by rows: T(n,k) (0<=k<=floor(n/2)) is the number of Delannoy paths of length n, having k ED's.

1, 3, 11, 2, 45, 18, 195, 120, 6, 873, 720, 90, 3989, 4110, 870, 20, 18483, 22806, 6930, 420, 86515, 124264, 49560, 5320, 70, 408105, 668520, 331128, 52920, 1890, 1936881, 3562830, 2111760, 456120, 29610, 252, 9238023, 18850590, 13020480, 3575880

Offset: 0

Author

Emeric Deutsch, Jul 20 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 n has 1+floor(n/2) terms. Row sums are the central Delannoy numbers (A001850). Column 0 yields A026375. Sum(k*T(n,k),k=0..floor(n/2))=2*A002695(n).

Examples

			T(2,1)=2 because we have NED and EDN.
Triangle begins:
1;
3;
11,2;
45,18;
195,120,6;
		

Links

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

Crossrefs

Cf. A001850, A026375, A002695.

Programs

Maple

R:=(1-z-sqrt(1-6*z+5*z^2-4*z^2*t))/2/z/(1-z+t*z): G:=1/(1-z-2*t*z^2*R-2*z*R+2*z^2*R): Gser:=simplify(series(G,z=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 12 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields sequence in triangular form

Formula

G.f.: 1/(1-z-2tz^2*R-2zR+2z^2*R), where R=[1-z-sqrt(1-6z+5z^2-4tz^2)]/[2z(1-z+tz)].