A266386 Sum over all Motzkin paths of length n of products over all peaks p of (x_p+n*y_p)/y_p, where x_p and y_p are the coordinates of peak p.

1, 1, 4, 11, 62, 243, 1575, 7721, 54985, 316407, 2427309, 15798261, 129072167, 927577835, 8008756470, 62499194297, 567017727805, 4747097031375, 45051331382395, 400942371431173, 3965769826314532, 37252002703698003, 382848953452815450, 3774255187367667473

Offset: 0

Author

Alois P. Heinz, Dec 28 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• Wikipedia, Motzkin number

Crossrefs

Main diagonal of A258306.

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, (x+k*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..30);

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, (x + k*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, 30}] (* Jean-François Alcover, Jun 10 2017, translated from Maple *)

Formula

a(n) = A258306(n,n).