A258172 Sum over all Dyck paths of semilength n of products over all peaks p of x_p, where x_p is the x-coordinate of peak p.

1, 1, 5, 40, 434, 5901, 95997, 1812525, 38875265, 932135347, 24678938063, 714385754446, 22428656766320, 758632387171075, 27489135956517315, 1061913384743418360, 43550536908458238570, 1889211624465639489675, 86406059558668152123975, 4154647501527354507485040

Offset: 0

Author

Alois P. Heinz, May 22 2015

Keywords

• nonn

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, Table of n, a(n) for n = 0..300
• Wikipedia, Lattice path

Crossrefs

Cf. A000108, A000698, A005411, A005412, A258173, A258174, A258175, A258176, A258177, A258178, A258179, A258180, A258181.

Programs

Maple

b:= proc(x, y, t) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false)*`if`(t, x, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

Mathematica

b[x_, y_, t_] := b[x, y, t] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False]*If[t, x, 1] + b[x - 1, y + 1, True]]];
a[n_] := b[2*n, 0, False];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)