A110183 Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n, having k (1,1)-steps on the lines y=x, y=x+1 and y=x-1.

1, 2, 1, 6, 6, 1, 22, 28, 12, 1, 90, 130, 80, 20, 1, 394, 616, 462, 180, 30, 1, 1806, 2982, 2538, 1270, 350, 42, 1, 8558, 14708, 13676, 8056, 2968, 616, 56, 1, 41586, 73698, 73176, 48392, 21608, 6188, 1008, 72, 1, 206098, 374224, 390926, 281948, 144512

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). First column yields the large Schroeder numbers (A006318). sum(k*T(n,k),k=0..n)=A110184(n)

Links

Table of n, a(n) for n=0..49.

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

Formula

G.f.: (1-2tz+z+Q)/[1-3z-3tz-tz^2+2t^2*z^2+(1-tz)Q], where Q=sqrt(1-6z+z^2).

Example

T(2,1)=6 because we have DNE, DEN, NED, END, NDE and EDN.
Triangle begins
1;
2,1;
6,6,1;
22,28,12,1;
90,130,80,20,1;

Maple

r:=(1-z-sqrt(1-6*z+z^2))/2/z: R:=1/(1-t*z-z*r): G:=1/(1-t*z-2*z*R): Gser:=simplify(series(G, z=0, 13)): P[0]:=1: for n from 1 to 10 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

Crossrefs

Cf. A001850, A006318, A110184.

Sequence in context: A104684 A060538 A260848 * A110098 A244888 A130561

Adjacent sequences: A110180 A110181 A110182 * A110184 A110185 A110186

Keyword

nonn,tabl

Author

Emeric Deutsch, Jul 14 2005

Status

approved