A292694 Sum over all Dyck paths of semilength n of products over all peaks p of (n*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p.

1, 2, 24, 764, 47650, 4953222, 776036520, 171140340632, 50569280587134, 19291547098210250, 9231053150452094896, 5414004448824367167444, 3819333773584571070766756, 3190486349393577447421521614, 3114480787139044226695876470000, 3512892958123523912923517986350000

Offset: 0

Author

Alois P. Heinz, Sep 20 2017

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Wikipedia, Lattice path

Crossrefs

Main diagonal of A258222.

Programs

Maple

b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
      `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
                 + b(x-1, y+1, true, k)  ))
    end:
a:= n-> b(2*n, 0, false, n):
seq(a(n), n=0..18);

Mathematica

b[x_, y_, t_, k_] := b[x, y, t, k] = If[y>x || y<0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k] If[t, (k*x + y)/y, 1] + b[x - 1, y + 1, True, k]]];
a[n_] := b[2n, 0, False, n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jun 02 2018, from Maple *)

Formula

a(n) = A258222(n,n).