A258312 Sum over all Motzkin paths of length 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, 2, 5, 14, 43, 141, 490, 1785, 6789, 26809, 109632, 462755, 2012441, 8997402, 41297927, 194306557, 936082502, 4612095475, 23219012907, 119328025012, 625545408219, 3342370197206, 18190297736313, 100768960522871, 567886743369378, 3253833477309093

Offset: 0

Author

Alois P. Heinz, May 25 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..800
• Wikipedia, Motzkin number

Crossrefs

Column k=0 of A258306 and A258307.

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, false)+b(x-1, y+1, true)))
    end:
a:= n-> b(n, 0, false):
seq(a(n), n=0..30);

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, False] + b[x-1, y+1, True]]];
a[n_] := b[n, 0, False];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 10 2017, translated from Maple *)