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

0,8

Links

Alois P. Heinz, Rows n = 0..200, flattened

Example

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;
  ...

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 *)

Crossrefs

Column k=0 gives A151346.

Row sums give A151404.

T(2n,n) gives A306813.

T(n+1,n-1) gives A001477.

T(n+2,n-1) gives A000217.

T(n+3,n-1) gives A000096.

Cf. A199915.

Sequence in context: A201076 A201079 A318601 * A241382 A049260 A273294

Adjacent sequences: A306811 A306812 A306813 * A306815 A306816 A306817

Keyword

nonn,tabl,walk

Author

Alois P. Heinz, Mar 11 2019

Status

approved