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).

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

0,8

Links

Alois P. Heinz, Rows n = 0..140, flattened

Example

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;

Maple

b:= proc(n, k, x, y) option remember;
     `if`(n<0 or x<0 or y<0 or n<x or n<abs(k-y), 0,
     `if`(n=0, 1, add (b(n-1, k, x+d[1], y+d[2]),
          d=[[1, 0], [0, 1], [0, -1], [-1, 1]])))
    end:
T:= (n, k)-> b(n, k, 0, 0):
seq(seq(T(n, k), k=0..n), n=0..12);

Mathematica

b[n_, k_, x_, y_] := b[n, k, x, y] = If[n<0 || x<0 || y<0 || n<x || n<Abs[k-y], 0, If[n == 0, 1, Sum[b[n-1, k, x+d[[1]], y+d[[2]]], {d, {{1, 0}, {0, 1}, {0, -1}, {-1, 1}}}]]]; T[n_, k_] := b[n, k, 0, 0]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 19 2015, after Alois P. Heinz *)

Crossrefs

Cf. A151346 (columns k=0, 1), A000217(n) = T(n+1,n), A151412 (row sums).

T(2n,n) gives A317782.

Cf. A306814.

Sequence in context: A205699 A109200 A158909 * A209557 A183759 A101477

Adjacent sequences: A199912 A199913 A199914 * A199916 A199917 A199918

Keyword

nonn,walk,tabl

Author

Alois P. Heinz, Nov 11 2011

Status

approved