A247749 Number T(n,k) of lattice paths from (0,0) to (n,0) that do not go below the x-axis or above the diagonal x=y, consist of steps u=(1,1), U=(1,3), H=(1,0), d=(1,-1) and D=(1,-3) for which the area below the path is k; triangle T(n,k), n>=0, read by rows.

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 2, 2, 0, 1, 1, 4, 6, 6, 6, 3, 4, 2, 1, 1, 1, 5, 10, 13, 15, 12, 14, 15, 9, 12, 5, 5, 1, 1, 1, 6, 15, 24, 32, 33, 37, 46, 40, 43, 34, 28, 23, 16, 10, 5, 2, 1, 1, 7, 21, 40, 61, 75, 88, 114, 122, 134, 137, 118, 127, 101, 99, 69, 68, 41, 38, 19, 17, 5, 5, 0, 1

Offset: 0

Author

Alois P. Heinz, Sep 23 2014

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
1;
1;
1, 1;
1, 2,  1;
1, 3,  3,  2,  2,  0,  1;
1, 4,  6,  6,  6,  3,  4,  2,  1,  1;
1, 5, 10, 13, 15, 12, 14, 15,  9, 12,  5,  5,  1,  1;
1, 6, 15, 24, 32, 33, 37, 46, 40, 43, 34, 28, 23, 16, 10, 5, 2, 1;
		

Links

• Alois P. Heinz, Rows n = 0..50, flattened

Crossrefs

Cf. A240904, A247748.

Programs

Maple

b:= proc(x, y) option remember; `if`(y<0 or x (p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);

Mathematica

b[x_, y_] := b[x, y] = If[y < 0 || x < y, 0, If[x == 0, 1,
     Expand[Sum[z^(y+j/2)*b[x-1, y+j], {j, {-1, -3, 0, 1, 3}}]]]];
T[n_] := CoefficientList[b[n, 0], z];
Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 29 2022, after Alois P. Heinz *)

Formula

Sum_{k>=0} T(n,k) = A240904(n).

Sum_{k>=1} k * T(n,k) = A247748(n).