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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Wikipedia, Motzkin number

Formula

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

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 *)

Crossrefs

Main diagonal of A258309.

Sequence in context: A222764 A058014 A290392 * A331013 A362283 A276912

Adjacent sequences: A261782 A261783 A261784 * A261786 A261787 A261788

Keyword

nonn

Author

Alois P. Heinz, Aug 31 2015

Status

approved