A258174 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, 1, 7, 84, 1486, 35753, 1111931, 43150593, 2035666985, 114412223081, 7538224510181, 574552299138202, 50096579094908148, 4949493445607316419, 549534510282406667069, 68071071679372210762156, 9347203754680124767253730, 1414740620049957735248175695

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..250
• Wikipedia, Lattice path

Crossrefs

Cf. A000108, A000698, A005411, A005412, A258172, A258173, 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*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 *)