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).

1, 2, 1, 6, 6, 1, 22, 30, 10, 1, 90, 146, 70, 14, 1, 394, 714, 430, 126, 18, 1, 1806, 3534, 2490, 938, 198, 22, 1, 8558, 17718, 14002, 6314, 1734, 286, 26, 1, 41586, 89898, 77550, 40054, 13338, 2882, 390, 30, 1, 206098, 461010, 426150, 244790, 94554, 24970, 4446, 510, 34, 1

Offset: 0

Author

Emeric Deutsch, Jul 11 2005

Keywords

• nonn
• tabl

Comments

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).

The row sums are the central Delannoy numbers (A001850).

Column 0 yields the large Schroeder numbers (A006318).

Column 1 yields A006320.

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).

Examples

			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).
Triangle begins:
   1;
   2,   1;
   6,   6,   1;
  22,  30,  10,   1;
  90, 146,  70,  14,   1;
		

Links

• T.-X. He and L. W. Shapiro, Fuss-Catalan matrices, their weighted sums, and stabilizer subgroups of the Riordan group, Lin. Alg. Applic. 532 (2017) 25-41, example p. 37.
• Robert A. Sulanke, Objects Counted by the Central Delannoy Numbers, Journal of Integer Sequences, Volume 6, 2003, Article 03.1.5.

Crossrefs

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

Programs

Maple

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

Mathematica

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}]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 21 2024, after Maple program *)

Formula

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;

T(n,n) = 1;

T(n,k) = 0 for k > n.

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).

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

T(n,k) = A033877(n-k+1, n+k+1). - Johannes W. Meijer, Sep 05 2013

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