A261785 Sum over all Motzkin paths of length 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, 1, 4, 13, 101, 571, 6735, 54713, 873019, 9274471, 187278048, 2460190261, 60205154959, 942541045811, 27121249048036, 492972449490417, 16312991079531595, 337650093459084079, 12633283010644517490, 293339323822142071021, 12245145846336974734339

Offset: 0

Author

Alois P. Heinz, Aug 31 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• Wikipedia, Motzkin number

Crossrefs

Main diagonal of A258309.

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, false, k) +b(x-1, y+1, true, k)))
    end:
a:= n-> b(n, 0, false, n):
seq(a(n), n=0..25);

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, False, k] + b[x - 1, y + 1, True, k]]];
a[n_] := b[n, 0, False, n];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 10 2017, translated from Maple *)

Formula

a(n) = A258309(n,n).