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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Wikipedia, Motzkin number

Formula

a(n) = A258306(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, (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 *)

Crossrefs

Main diagonal of A258306.

Sequence in context: A242749 A303955 A114053 * A134823 A000880 A006551

Adjacent sequences: A266383 A266384 A266385 * A266387 A266388 A266389

Keyword

nonn

Author

Alois P. Heinz, Dec 28 2015

Status

approved