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%I #7 Apr 04 2019 08:15:35
%S 1,2,1,8,4,1,36,20,6,1,172,104,36,8,1,852,552,212,56,10,1,4324,2968,
%T 1236,368,80,12,1,22332,16104,7164,2336,580,108,14,1,116876,87976,
%U 41372,14512,3980,856,140,16,1,618084,483192,238356,88848,26372,6312,1204,176
%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 line y=x (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).
%C First column yields A109980.
%C sum(k*T(n,k),k=0..n) = A001109(n).
%H Robert A. Sulanke, <a href="https://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.: [tz-z+sqrt(1-6z+z^2)]/(1-6z+2tz^2-t^2*z^2).
%e T(2,1)=4 because we have DNE, DEN, NED and END.
%e Triangle begins:
%e 1;
%e 2,1;
%e 8,4,1;
%e 36,20,6,1;
%p G:=(t*z-z+sqrt(1-6*z+z^2))/(1-6*z+2*t*z^2-t^2*z^2): 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, A109980, A001109.
%K nonn,tabl
%O 0,2
%A _Emeric Deutsch_, Jul 06 2005
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