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%I #16 Jul 23 2017 10:55:33
%S 1,2,1,6,6,1,22,30,10,1,90,146,70,14,1,394,714,430,126,18,1,1806,3534,
%T 2490,938,198,22,1,8558,17718,14002,6314,1734,286,26,1,41586,89898,
%U 77550,40054,13338,2882,390,30,1,206098,461010,426150,244790,94554
%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 (i.e., E 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 The row sums are the central Delannoy numbers (A001850).
%C Column 0 yields the large Schroeder numbers (A006318).
%C Column 1 yields A006320.
%C Column k has g.f. z^k*R^(2*k+1), where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318).
%H T.-X. He, L. W. Shapiro, <a href="http://dx.doi.org/10.1016/j.laa.2017.06.025">Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group</a>, Lin. Alg. Applic. 532 (2017) 25-41, example p 37.
%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 T(n,k) = ((2*k+1)/(n-k))*Sum_{j=0..n-k} binomial(n-k, j)*binomial(n+k+j, n-k-1) for k < n;
%F T(n,n) = 1;
%F T(n,k) = 0 for k > n.
%F G.f.: R/(1 - t*z*R^2), where R = 1 + z*R + z*R^2 is the g.f. of the large Schroeder numbers (A006318).
%F Sum_{k=0..n} k*T(n,k) = A110099(n).
%F T(n,k) = A033877(n-k+1, n+k+1). - _Johannes W. Meijer_, Sep 05 2013
%e T(2, 1) = 6 because we have DN(E), N(E)D, N(E)EN, ND(E), NNE(E) and ENN(E) (the return E steps are shown between parentheses).
%e Triangle begins:
%e 1;
%e 2, 1;
%e 6, 6, 1;
%e 22, 30, 10, 1;
%e 90, 146, 70, 14, 1;
%p T := proc(n, k) if k=n then 1 else ((2*k+1)/(n-k))*sum(binomial(n-k,j)*binomial(n+k+j,n-k-1),j=0..n-k) fi end: for n from 0 to 10 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
%Y Cf. A001850, A006318, A006320, A033877, A110099, A110107.
%K nonn,tabl
%O 0,2
%A _Emeric Deutsch_, Jul 11 2005
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