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

0,6

Links

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

Formula

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

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

Example

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;

Maple

b:= proc(x, y) option remember; `if`(y<0 or x<y, 0, `if`(x=0, 1,
      expand(add(z^(y+j/2)*b(x-1, y+j), j=[-1, -3, 0, 1, 3]))))
    end:
T:= n-> (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 *)

Crossrefs

Cf. A240904, A247748.

Sequence in context: A131336 A052253 A271453 * A247367 A305321 A346798

Adjacent sequences: A247746 A247747 A247748 * A247750 A247751 A247752

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 23 2014

Status

approved