A258175 Sum over all Dyck paths of semilength n of products over all peaks p of x_p+y_p, where x_p and y_p are the coordinates of peak p.

1, 2, 12, 114, 1448, 22770, 424164, 9095450, 220023184, 5914998594, 174682531260, 5614908340866, 194967208057272, 7267467723747218, 289270983756577620, 12239218862861690250, 548301077168477951520, 25918121712918957399426, 1288797080051656060595820

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..350
• Wikipedia, Lattice path

Crossrefs

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, 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+y, 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 + y, 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 *)