A110098 - OEIS

%I #28 Oct 10 2024 06:55:08

%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 77550,40054,13338,2882,390,30,1,206098,461010,426150,244790,94554,24970,4446,510,34,1

%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 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 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

%F 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 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

%t 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 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 21 2024, after Maple program *)

%Y Cf. A001850, A063007, A006318, A006320, A033877, A110099, A110107.

%K nonn,tabl,changed

%O 0,2

%A _Emeric Deutsch_, Jul 11 2005