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A110107 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n, having k return steps to the line y = x from the line y = x+1 or from the line y = x-1 (i.e., E steps from the line y = x+1 to the line y = x or N steps from the line y = x-1 to the line y = x).

%I A110107 #15 Jul 23 2017 12:18:27
%S A110107 1,1,2,1,8,4,1,26,28,8,1,88,136,80,16,1,330,600,512,208,32,1,1360,
%T A110107 2636,2768,1648,512,64,1,6002,11892,14024,10544,4832,1216,128,1,27760,
%U A110107 55376,69728,60768,35712,13312,2816,256,1,132690,265200,347072,332768,231232
%N A110107 Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n, having k return steps to the line y = x from the line y = x+1 or from the line y = x-1 (i.e., E steps from the line y = x+1 to the line y = x or N steps from the line y = x-1 to the line y = x).
%C A110107 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 A110107 Row sums are the central Delannoy numbers (A001850).
%H A110107 Alois P. Heinz, <a href="/A110107/b110107.txt">Rows n = 0..140, flattened</a>
%H A110107 R. A. Sulanke, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL6/Sulanke/delannoy.html">Objects counted by the central Delannoy numbers</a>, J. Integer Seq. 6 (2003), no. 1, Article 03.1.5.
%F A110107 Sum_{k=0..n} k*T(n,k) = 2*A110099(n).
%F A110107 G.f.: 1/(1 - z - 2tzR), where R = 1 + zR+ z R^2 is the g.f. of the large Schroeder numbers (A006318).
%e A110107 T(2,1) = 8 because we have DN(E), DE(N), N(E)D, ND(E), NNE(E), E(N)D, ED(N) and EEN(N) (the return E or N steps are shown between parentheses).
%e A110107 Triangle begins:
%e A110107   1;
%e A110107   1,   2;
%e A110107   1,   8,   4;
%e A110107   1,  26,  28,   8;
%e A110107   1,  88, 136,  80,  16;
%p A110107 R:=(1-z-sqrt(1-6*z+z^2))/2/z: G:=1/(1-z-2*t*z*R): Gser:=simplify(series(G,z=0,12)): P[0]:=1: for n from 1 to 9 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 A110107 Cf. A001850, A110098, A110099.
%K A110107 nonn,tabl
%O A110107 0,3
%A A110107 _Emeric Deutsch_, Jul 11 2005