cp's OEIS Frontend

A110098 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).

%I A110098 #28 Oct 10 2024 06:55:08
%S A110098 1,2,1,6,6,1,22,30,10,1,90,146,70,14,1,394,714,430,126,18,1,1806,3534,
%T A110098 2490,938,198,22,1,8558,17718,14002,6314,1734,286,26,1,41586,89898,
%U A110098 77550,40054,13338,2882,390,30,1,206098,461010,426150,244790,94554,24970,4446,510,34,1
%N A110098 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 A110098 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 A110098 The row sums are the central Delannoy numbers (A001850).
%C A110098 Column 0 yields the large Schroeder numbers (A006318).
%C A110098 Column 1 yields A006320.
%C A110098 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 A110098 T.-X. He and 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 A110098 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 A110098 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 A110098 T(n,n) = 1;
%F A110098 T(n,k) = 0 for k > n.
%F A110098 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 A110098 Sum_{k=0..n} k*T(n,k) = A110099(n).
%F A110098 T(n,k) = A033877(n-k+1, n+k+1). - _Johannes W. Meijer_, Sep 05 2013
%F A110098 It appears that this triangle equals M * N^(-1), where M is the lower triangular array A063007 and N = ( (-1)^(n+k)* binomial(n, k)*binomial(n+k, k) )n,k >= 0 is a signed version of A063007. - _Peter Bala_, Oct 07 2024
%e A110098 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 A110098 Triangle begins:
%e A110098    1;
%e A110098    2,   1;
%e A110098    6,   6,   1;
%e A110098   22,  30,  10,   1;
%e A110098   90, 146,  70,  14,   1;
%p A110098 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
%t A110098 T[n_, k_] := If[k == n, 1, ((2*k+1)/(n-k))*Sum[Binomial[n-k, j]*Binomial[n+k+j, n-k-1], {j, 0, n-k}]];
%t A110098 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 21 2024, after Maple program *)
%Y A110098 Cf. A001850, A063007, A006318, A006320, A033877, A110099, A110107.
%K A110098 nonn,tabl
%O A110098 0,2
%A A110098 _Emeric Deutsch_, Jul 11 2005