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%I #16 Jul 03 2017 02:21:49
%S 1,1,1,1,2,1,1,3,3,1,1,4,8,4,1,1,5,15,15,5,1,1,6,24,30,24,6,1,1,7,35,
%T 54,54,35,7,1,1,8,48,89,138,89,48,8,1,1,9,63,137,281,281,137,63,9,1,1,
%U 10,80,200,507,562,507,200,80,10,1,1,11,99,280,844
%N Triangular array T read by rows: T(n,0)=T(n,n)=1 for n >= 0; for n >= 2 and 1<=k<=n-1, T(n,k)=T(n-1,k-1)+T(n-2,k-1)+T(n-1,k) if k or n-k is of form 2h for h=1,2,...,[ n/4 ], else T(n,k)=T(n-1,k-1)+T(n-1,k).
%F T(n, k) = number of paths from (0, 0) to (n-k, k) in directed graph having vertices (i, j) and edges (i, j)-to-(i+1, j) and (i, j)-to-(i, j+1) for i, j >= 0 and edges (i, j)-to-(i+1, j+1) for i odd and j >= i and for j odd and i >= j.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 3, 1;
%e 1, 4, 8, 4, 1;
%e 1, 5, 15, 15, 5, 1;
%e ...
%t T[_, 0] = 1; T[n_, n_] = 1; T[n_, k_] /; EvenQ[k] && 1 <= k/2 <= Floor[n/4] || EvenQ[n-k] && 1 <= (n-k)/2 <= Floor[n/4] := T[n, k] = T[n-1, k-1] + T[n-2, k-1] + T[n-1, k]; T[n_, k_] := T[n, k] = T[n-1, k-1] + T[n-1, k];
%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 02 2017 *)
%K nonn,tabl
%O 1,5
%A _Clark Kimberling_
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