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A096588 a(n)=number of Catalan knight paths in Quadrant I from (0,0) to points on the vertical line x=n. A Catalan knight moves (2 right and 1 up) or (1 right and 1 down).

%I A096588 #21 Jul 09 2025 04:23:50
%S A096588 1,1,3,6,16,38,99,248,646,1659,4342,11307,29740,78115,206349,545156,
%T A096588 1445332,3834559,10197168,27140709,72357778,193076677,515843630,
%U A096588 1379308111,3691755414,9888374480,26507373732,71103941488,190859621124
%N A096588 a(n)=number of Catalan knight paths in Quadrant I from (0,0) to points on the vertical line x=n. A Catalan knight moves (2 right and 1 up) or (1 right and 1 down).
%H A096588 Paolo Xausa, <a href="/A096588/b096588.txt">Table of n, a(n) for n = 0..1000</a>
%H A096588 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). Also arXiv:2206.12087, Jan 2023.
%F A096588 Row sums of array in A096587.
%e A096588 Rows of array T(n,k) in A096587:
%e A096588   1
%e A096588   0 0 1
%e A096588   1 1 0 0 1
%e A096588   0 1 2 2 0 0 1
%e A096588   ...
%e A096588 so a(3)=T(3,0)+T(3,1)+...+T(3,6)=6.
%t A096588 A096588list[nmax_]:=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[Sum[T[n,k],{k,0,2n}],{n,0,nmax}]];A096588list[50] (* _Paolo Xausa_, May 22 2023 *)
%Y A096588 Cf. A096587.
%K A096588 nonn
%O A096588 0,3
%A A096588 _Clark Kimberling_, Jun 28 2004