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
Examples
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;
Links
- Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.
Programs
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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
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).
Comments