A110169 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n that start with exactly k (1,1) steps.

1, 2, 1, 10, 2, 1, 50, 10, 2, 1, 258, 50, 10, 2, 1, 1362, 258, 50, 10, 2, 1, 7306, 1362, 258, 50, 10, 2, 1, 39650, 7306, 1362, 258, 50, 10, 2, 1, 217090, 39650, 7306, 1362, 258, 50, 10, 2, 1, 1196834, 217090, 39650, 7306, 1362, 258, 50, 10, 2, 1, 6634890, 1196834

Offset: 0

Author

Emeric Deutsch, Jul 14 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). Column 0 yields A110170 (first differences of the central Delannoy numbers). sum(k*T(n,k),k=0..n)=A089165(n-1) (n>=1; partial sums of the central Delannoy numbers).

Examples

			T(3,2)=2 because we have DDNE and DDEN.
Triangle starts:
1;
2,1;
10,2,1;
50,10,2,1;
258,50,10,2,1;
		

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, A110170, A089165.

Programs

Maple

with(orthopoly): S:=proc(n,k) if k
				

Formula

T(n, k) = A001850(n-k)-A001850(n-k-1) for k

T(n, k) = P_{n-k}(3)-P_{n-k-1}(3) for k

G.f.: (1-z)/((1-t*z)*sqrt(1-6*z+z^2)).

Extensions

Keyword tabf changed to tabl by Michel Marcus, Apr 09 2013

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