A258176 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, 142, 9354, 2503597, 3260627607, 24105227716863, 1028836978599566213, 290383808553140390346475, 511963364817949502725911280781, 6704846980724405836568589845161191576, 584709361918378923208855262622537662297053728

Offset

0,3

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..45

Wikipedia, Lattice path

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..15);

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, 15}] (* Jean-François Alcover, Apr 23 2016, translated from Maple *)

Crossrefs

Cf. A000108, A000698, A005411, A005412, A258172, A258173, A258174, A258175, A258177, A258178, A258179, A258180, A258181.

Sequence in context: A179569 A342812 A082157 * A333088 A286398 A104240

Adjacent sequences: A258173 A258174 A258175 * A258177 A258178 A258179

Keyword

nonn

Author

Alois P. Heinz, May 22 2015

Status

approved