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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). 3

%I

%S 1,1,2,1,8,4,1,26,28,8,1,88,136,80,16,1,330,600,512,208,32,1,1360,

%T 2636,2768,1648,512,64,1,6002,11892,14024,10544,4832,1216,128,1,27760,

%U 55376,69728,60768,35712,13312,2816,256,1,132690,265200,347072,332768,231232

%N 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 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).

%H Alois P. Heinz, <a href="/A110107/b110107.txt">Rows n = 0..140, flattened</a>

%H 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 Sum_{k=0..n} k*T(n,k) = 2*A110099(n).

%F G.f.: 1/(1 - z - 2tzR), where R = 1 + zR+ z R^2 is the g.f. of the large Schroeder numbers (A006318).

%e 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 Triangle begins:

%e 1;

%e 1, 2;

%e 1, 8, 4;

%e 1, 26, 28, 8;

%e 1, 88, 136, 80, 16;

%p 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 Cf. A001850, A110098, A110099.

%K nonn,tabl

%O 0,3

%A _Emeric Deutsch_, Jul 11 2005

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Last modified June 21 18:50 EDT 2021. Contains 345365 sequences. (Running on oeis4.)