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

0,2

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).

Links

Table of n, a(n) for n=0..56.

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

Formula

T(n, k) = A001850(n-k)-A001850(n-k-1) for k<n; T(n, n)=1.

T(n, k) = P_{n-k}(3)-P_{n-k-1}(3) for k<n; T(n, n)=1, where P_j is j-th Legendre polynomial.

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

Example

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;

Maple

with(orthopoly): S:=proc(n, k) if k<n then P(n-k, 3)-P(n-k-1, 3) elif k=n then 1 else 0 fi end: for n from 0 to 10 do seq(S(n, k), k=0..n) od; # yields sequence in triangular form

Crossrefs

Cf. A001850, A110170, A089165.

Sequence in context: A132995 A114692 A112691 * A144274 A144275 A247236

Adjacent sequences: A110166 A110167 A110168 * A110170 A110171 A110172

Keyword

nonn,tabl

Author

Emeric Deutsch, Jul 14 2005

Extensions

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

Status

approved