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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

%I

%S 1,2,1,6,6,1,22,28,12,1,90,130,80,20,1,394,616,462,180,30,1,1806,2982,

%T 2538,1270,350,42,1,8558,14708,13676,8056,2968,616,56,1,41586,73698,

%U 73176,48392,21608,6188,1008,72,1,206098,374224,390926,281948,144512

%N 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.

%C 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).

%C 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)

%H Robert A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects Counted by the Central Delannoy Numbers</a>, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

%F 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).

%e T(2,1)=6 because we have DNE, DEN, NED, END, NDE and EDN.

%e Triangle begins

%e 1;

%e 2,1;

%e 6,6,1;

%e 22,28,12,1;

%e 90,130,80,20,1;

%p 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

%Y Cf. A001850, A006318, A110184.

%K nonn,tabl

%O 0,2

%A _Emeric Deutsch_, Jul 14 2005

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Last modified May 17 06:30 EDT 2021. Contains 343965 sequences. (Running on oeis4.)