A199915
Triangle read by rows: T(n,k) is the number 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).
Original entry on oeis.org
1, 0, 1, 1, 1, 1, 1, 2, 3, 1, 2, 7, 5, 6, 1, 7, 10, 21, 14, 10, 1, 10, 38, 48, 51, 35, 15, 1, 38, 89, 135, 168, 120, 76, 21, 1, 89, 229, 441, 458, 474, 281, 147, 28, 1, 229, 752, 1121, 1604, 1475, 1188, 637, 260, 36, 1, 752, 1873, 3692, 4772, 5100, 4329, 2800, 1366, 429, 45, 1
Offset: 0
T(4,2) = 5: ((1,0),(1,0),(-1,1),(-1,1)); ((1,0),(-1,1),(1,0),(-1,1)); ((0,1),(0,1),(0,1),(0,-1)); ((0,1),(0,1),(0,-1),(0,1)); ((0,1),(0,-1),(0,1),(0,1)).
Triangle begins:
1;
0, 1;
1, 1, 1;
1, 2, 3, 1;
2, 7, 5, 6, 1;
7, 10, 21, 14, 10, 1;
10, 38, 48, 51, 35, 15, 1;
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b:= proc(n, k, x, y) option remember;
`if`(n<0 or x<0 or y<0 or n b(n, k, 0, 0):
seq(seq(T(n, k), k=0..n), n=0..12);
-
b[n_, k_, x_, y_] := b[n, k, x, y] = If[n<0 || x<0 || y<0 || nJean-François Alcover, Jan 19 2015, after Alois P. Heinz *)
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.
Original entry on oeis.org
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
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;
...
-
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);
-
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 *)
Showing 1-2 of 2 results.