A258179 Sum over all Dyck paths of semilength n of products over all peaks p of y_p^2, where y_p is the y-coordinate of peak p.

1, 1, 5, 34, 312, 3649, 52161, 889843, 17796555, 411120395, 10838039407, 322752018060, 10762432731362, 398802951148255, 16312276452291935, 732189190349581890, 35876807697443520000, 1910107567584518883891, 110035833179472385285367, 6832792252684597270659486

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, A258172, A258173, A258174, A258175, A258176, A258177, A258178, 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, y^2, 1) +
                   b(x-1, y+1, true)  ))
    end:
a:= n-> b(2*n, 0, false):
seq(a(n), n=0..20);

Mathematica

nmax = 20; Clear[g]; g[nmax+1] = 1; g[k_] := g[k] = 1 - x/( (k+2)^2*x - 1/g[k+1]); CoefficientList[Series[g[0], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 20 2015, after Sergei N. Gladkovskii *)

Formula

G.f.: T(0), where T(k) = 1 - x/( (k+2)^2*x - 1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Aug 20 2015