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%I #30 Jul 16 2019 21:58:39
%S 1,1,1,1,2,1,1,3,3,1,1,5,6,4,1,1,6,11,10,5,1,1,7,22,21,15,6,1,1,8,29,
%T 43,36,21,7,1,1,9,37,94,79,57,28,8,1,1,10,46,131,173,136,85,36,9,1,1,
%U 11,56,177,398,309,221,121,45,10,1,1,12,67,233,575,707,530,342,166,55,11,1
%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 n is even and k=(n-2)/2, otherwise T(n,k) = T(n-1,k-1) + T(n-1,k).
%C T(n, k) is the 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, i+2)-to-(i+1, i+3) for i >= 0.
%H G. C. Greubel, <a href="/A026736/b026736.txt">Rows n = 0..100 of triangle, flattened</a>
%e Triangle begins
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 3, 1;
%e 1, 5, 6, 4, 1;
%e 1, 6, 11, 10, 5, 1;
%e 1, 7, 22, 21, 15, 6, 1;
%e 1, 8, 29, 43, 36, 21, 7, 1;
%e 1, 9, 37, 94, 79, 57, 28, 8, 1;
%e 1, 10, 46, 131, 173, 136, 85, 36, 9, 1;
%e 1, 11, 56, 177, 398, 309, 221, 121, 45, 10, 1;
%e 1, 12, 67, 233, 575, 707, 530, 342, 166, 55, 11, 1;
%e ...
%t T[_, 0] = T[n_, n_] = 1; T[n_, k_] := T[n, k] = If[EvenQ[n] && k == (n-2)/2, T[n-1, k-1] + T[n-2, k-1] + T[n-1, k], T[n-1, k-1] + T[n-1, k]];
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 22 2018 *)
%o (PARI)
%o T(n,k) = if(k==n || k==0, 1, if((n%2)==0 && k==(n-2)/2, T(n-1, k-1) + T(n-2, k-1) + T(n-1, k), T(n-1, k-1) + T(n-1, k) ));
%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jul 16 2019
%o (Sage)
%o def T(n, k):
%o if (k==0 or k==n): return 1
%o elif (mod(n,2)==0 and k==(n-2)/2): return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k)
%o else: return T(n-1, k-1) + T(n-1, k)
%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jul 16 2019
%o (GAP)
%o T:= function(n,k)
%o if k=0 or k=n then return 1;
%o elif (n mod 2)=0 and k=Int((n-2)/2) then return T(n-1, k-1) + T(n-2, k-1) + T(n-1, k);
%o else return T(n-1, k-1) + T(n-1, k);
%o fi;
%o end;
%o Flat(List([0..12], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Jul 16 2019
%Y Row sums give A026743.
%Y T(2n,n) gives A026737(n) or A111279(n+1).
%K nonn,tabl,walk
%O 0,5
%A _Clark Kimberling_
%E Offset corrected by _Alois P. Heinz_, Jul 23 2018
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