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

0,5

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.

Links

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

Wikipedia, Lattice path

Formula

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

Example

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, ...

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 *)

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.

Sequence in context: A262157 A090447 A241186 * A112324 A061531 A368093

Adjacent sequences: A258219 A258220 A258221 * A258223 A258224 A258225

Keyword

nonn,tabl

Author

Alois P. Heinz, May 23 2015

Status

approved