cp's OEIS Frontend

A096608 Triangle read by rows: T(n,k)=number of Catalan knight paths in right half-plane from (0,0) to (n,k), for 0 <= k <= 2n, n >= 0. (See A096587 for the definition of a Catalan knight.)

%I A096608 #39 Feb 13 2024 13:55:47
%S A096608 1,0,0,1,2,1,0,0,1,0,2,3,2,0,0,1,8,6,1,3,4,3,0,0,1,6,12,16,12,3,4,5,4,
%T A096608 0,0,1,44,33,18,21,27,20,6,5,6,5,0,0,1,60,76,95,72,40,34,41,30,10,6,7,
%U A096608 6,0,0,1,256,210,154,155,177,135,75,52,58,42,15,7,8,7,0,0,1,460,520,581,480
%N A096608 Triangle read by rows: T(n,k)=number of Catalan knight paths in right half-plane from (0,0) to (n,k), for 0 <= k <= 2n, n >= 0. (See A096587 for the definition of a Catalan knight.)
%H A096608 Paolo Xausa, <a href="/A096608/b096608.txt">Table of n, a(n) for n = 0..9999</a> (rows 0..99 of triangle, flattened)
%H A096608 Jean-Luc Baril, Nathanaël Hassler, Sergey Kirgizov, and José L. Ramírez, <a href="https://arxiv.org/abs/2402.04851">Grand zigzag knight's paths</a>, arXiv:2402.04851 [math.CO], 2024.
%H A096608 Rémy Sigrist, <a href="/A355320/a355320.png">Representation of the odd terms for n = 0..2^11</a>
%F A096608 T(0, 0) = 1, T(0, 1) = 0, T(0, 2) = 0; T(1, 0) = 0, T(1, 1) = 0, T(1, 2) = 1.
%F A096608 For n >= 2, T(n, 0) = 2*T(n-2, 1) + 2*T(n-1, 2); T(n, 1) = T(n-2, 0) + T(n-2, 2) + T(n-1, 3) + T(n-1, 1); for 2 <= k <= 2n, T(n, k) = T(n-2, k-1) + T(n-2, k+1) + T(n-1, k-2) + T(n-1, k+2).
%F A096608 T(n, 0) + 2*Sum_{k = 1..2*n} T(n, k) = A002605(k). - _Rémy Sigrist_, Jun 29 2022
%e A096608 Rows:
%e A096608   1;
%e A096608   0, 0, 1;
%e A096608   2, 1, 0, 0, 1;
%e A096608   0, 2, 3, 2, 0, 0, 1;
%e A096608 T(3,2) counts these paths:
%e A096608   (0,0)-(1,-2)-(2,0)-(3,2);
%e A096608   (0,0)-(1,2)-(2,0)-(3,2);
%e A096608   (0,0)-(1,2)-(2,4)-(3,2).
%t A096608 A096608[rowmax_]:=Module[{T},T[0,0]=1;T[n_,k_]:=T[n,k]=If[k<=2n,T[n-1,Abs[k-2]]+T[n-2,Abs[k-1]]+T[n-1,k+2]+T[n-2,k+1],0];Table[T[n,k],{n,0,rowmax},{k,0,2n}]]; A096608[10] (* Generates 11 rows *) (* _Paolo Xausa_, May 09 2023 *)
%o A096608 (PARI) row(n) = { my (rr=0, r=1); for (k=1, n, [rr, r]=[r, r*(1+'X^4)+rr*('X^3+'X^5)]); Vec(r)[1+2*n..1+4*n] } \\ _Rémy Sigrist_, Jun 29 2022
%Y A096608 Cf. A096587, A096609, A096610, A096611, A096612, A355320.
%K A096608 nonn,tabf
%O A096608 0,5
%A A096608 _Clark Kimberling_, Jun 29 2004
%E A096608 Offset changed to 0 by _Rémy Sigrist_, Jun 29 2022