A096587 Triangle read by rows: T(n,k)=number of Catalan knight paths in Quadrant I from (0,0) to (n,k), for 0 <= k <= 2*n, n >= 0. A Catalan knight moves (1 right and 2 up) or (1 right and 2 down) or (2 right and 1 up) or (2 right and 1 down).

1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 2, 2, 0, 0, 1, 3, 3, 1, 2, 3, 3, 0, 0, 1, 2, 4, 9, 8, 3, 3, 4, 4, 0, 0, 1, 12, 12, 10, 11, 18, 15, 6, 4, 5, 5, 0, 0, 1, 14, 22, 42, 39, 27, 22, 30, 24, 10, 5, 6, 6, 0, 0, 1, 54, 61, 64, 72, 98, 87, 56, 38, 45, 35, 15, 6, 7, 7, 0, 0, 1, 86, 128, 213, 217, 181, 167

Offset: 0

Author

Clark Kimberling, Jun 28 2004

Keywords

• nonn
• tabf

Examples

			Rows:
  1
  0 0 1
  1 1 0 0 1
  0 1 2 2 0 0 1
  ...
T(3,2) counts these paths: (0,0)-(1,2)-(2,0)-(3,2) and (0,0)-(1,2)-(2,4)-(3,2).
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..9999 (rows 0..99 of triangle, flattened)
• Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).

Crossrefs

Cf. A005220 (column 0), A005221 (column 1), A096588, A096608.

Programs

Mathematica

A096587list[rowmax_]:=Module[{T},T[0,0]=1;T[n_,k_]:=T[n,k]=If[0<=k<=2n,T[n-1,k-2]+T[n-2,k-1]+T[n-1,k+2]+T[n-2,k+1],0];Table[T[n,k],{n,0,rowmax},{k,0,2n}]];A096587list[10] (* Generates 11 rows *) (* Paolo Xausa, May 22 2023 *)

Formula

T(0, 0) = 1; T(1, 2) = 1; for n >= 2, T(n, 0) = T(n-2, 1)+T(n-1, 2), T(n, 1) = T(n-2, 0)+T(n-2, 2)+T(n-1, 3); for k >= 2, T(n, k) = T(n-2, k-1)+T(n-2, k+1)+T(n-1, k-2)+T(n-1, k+2).

Extensions

Offset changed to 0 by Paolo Xausa, May 22 2023