A096587 - OEIS

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).

%I #16 May 28 2023 11:31:16

%S 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,

%T 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,

%U 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

%N 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).

%H Paolo Xausa, <a href="/A096587/b096587.txt">Table of n, a(n) for n = 0..9999</a> (rows 0..99 of triangle, flattened)

%H Jean-Luc Baril and José L. Ramírez, <a href="http://jl.baril.u-bourgogne.fr/knight.pdf">Knight's paths towards Catalan numbers</a>, Univ. Bourgogne Franche-Comté (2022).

%F 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).

%e Rows:

%e 1

%e 0 0 1

%e 1 1 0 0 1

%e 0 1 2 2 0 0 1

%e ...

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

%t 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 *)

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

%K nonn,tabf

%O 0,12

%A _Clark Kimberling_, Jun 28 2004

%E Offset changed to 0 by _Paolo Xausa_, May 22 2023