A306814 Number T(n,k) of n-step paths from (0,0) to (0,k) that stay in the first quadrant (but may touch the axes) consisting of steps (-1,0), (0,1), (0,-1) and (1,-1); triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 2, 3, 3, 0, 1, 7, 5, 6, 4, 0, 1, 10, 23, 9, 10, 5, 0, 1, 38, 35, 51, 14, 15, 6, 0, 1, 89, 131, 84, 94, 20, 21, 7, 0, 1, 229, 355, 309, 168, 155, 27, 28, 8, 0, 1, 752, 874, 947, 608, 300, 237, 35, 36, 9, 0, 1, 1873, 3081, 2292, 2075, 1070, 495, 343, 44, 45, 10, 0, 1
Offset: 0
Examples
T(4,2) = 3:
[(0,0), (0,1), (0,0), (0,1), (0,2)],
[(0,0), (0,1), (0,2), (0,1), (0,2)],
[(0,0), (0,1), (0,2), (0,3), (0,2)].
Triangle T(n,k) begins:
1;
0, 1;
1, 0, 1;
1, 2, 0, 1;
2, 3, 3, 0, 1;
7, 5, 6, 4, 0, 1;
10, 23, 9, 10, 5, 0, 1;
38, 35, 51, 14, 15, 6, 0, 1;
89, 131, 84, 94, 20, 21, 7, 0, 1;
229, 355, 309, 168, 155, 27, 28, 8, 0, 1;
752, 874, 947, 608, 300, 237, 35, 36, 9, 0, 1;
...
Links
- Alois P. Heinz, Rows n = 0..200, flattened
Crossrefs
Programs
-
Maple
b:= proc(n, x, y) option remember; `if`(min(n, x, y, n-x-y)<0, 0, `if`(n=0, 1, add(b(n-1, x-d[1], y-d[2]), d=[[-1, 0], [0, 1], [0, -1], [1, -1]]))) end: T:= (n, k)-> b(n, 0, k): seq(seq(T(n, k), k=0..n), n=0..12); -
Mathematica
b[n_, x_, y_] := b[n, x, y] = If[Min[n, x, y, n - x - y] < 0, 0, If[n == 0, 1, Sum[b[n - 1, x - d[[1]], y - d[[2]]], {d, {{-1, 0}, {0, 1}, {0, -1}, {1, -1}}}]]]; T[n_, k_] := b[n, 0, k]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 14 2020, after Maple *)