VOOZH about

URL: https://oeis.org/A247749

⇱ A247749 - OEIS

login
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.
3
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
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
Sequence in context: A131336 A052253 A271453 * A247367 A305321 A346798
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Sep 23 2014
STATUS
approved