A258222 A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (k*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 2, 2, 1, 3, 10, 5, 1, 4, 24, 74, 14, 1, 5, 44, 297, 706, 42, 1, 6, 70, 764, 4896, 8162, 132, 1, 7, 102, 1565, 17924, 100278, 110410, 429, 1, 8, 140, 2790, 47650, 527844, 2450304, 1708394, 1430, 1, 9, 184, 4529, 104454, 1831250, 18685164, 69533397, 29752066, 4862

Offset: 0

Author

Alois P. Heinz, May 23 2015

Keywords

• nonn
• tabl

Comments

A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.

Examples

			Square array A(n,k) begins:
:  1,    1,      1,      1,       1,       1, ...
:  1,    2,      3,      4,       5,       6, ...
:  2,   10,     24,     44,      70,     102, ...
:  5,   74,    297,    764,    1565,    2790, ...
: 14,  706,   4896,  17924,   47650,  104454, ...
: 42, 8162, 100278, 527844, 1831250, 4953222, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• Wikipedia, Lattice path

Crossrefs

Columns k=0-1 give: A000108, A000698(n+1).

Rows n=0-2 give: A000012, A000027(k+1), A049450(k+1).

Main diagonal gives A292694.

Cf. A258219, A258223.

Programs

Maple

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
                 + b(x-1, y+1, true, k)  ))
    end:
A:= (n, k)-> b(2*n, 0, false, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1] + b[x - 1, y + 1, True, k]]];
A [n_, k_] := b[2*n, 0, False, k];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

Formula

A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258223(n,i).