A258306 A(n,k) is the sum over all Motzkin paths of length n of products over all peaks p of (x_p+k*y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 7, 14, 1, 1, 5, 9, 23, 43, 1, 1, 6, 11, 34, 71, 141, 1, 1, 7, 13, 47, 105, 255, 490, 1, 1, 8, 15, 62, 145, 411, 911, 1785, 1, 1, 9, 17, 79, 191, 615, 1496, 3535, 6789, 1, 1, 10, 19, 98, 243, 873, 2269, 6169, 13903, 26809

Offset: 0

Author

Alois P. Heinz, May 25 2015

Keywords

• nonn
• tabl

Examples

			Square array A(n,k) begins:
:   1,   1,   1,   1,   1,    1,    1, ...
:   1,   1,   1,   1,   1,    1,    1, ...
:   2,   3,   4,   5,   6,    7,    8, ...
:   5,   7,   9,  11,  13,   15,   17, ...
:  14,  23,  34,  47,  62,   79,   98, ...
:  43,  71, 105, 145, 191,  243,  301, ...
: 141, 255, 411, 615, 873, 1191, 1575, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..140, flattened
• Wikipedia, Motzkin number

Crossrefs

Columns k=0-1 give: A258312, A140456(n+2).

Main diagonal gives A266386.

Cf. A258307, 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, (x+k*y)/y, 1)
                  +b(x-1, y, false, k) +b(x-1, y+1, true, k)))
    end:
A:= (n, k)-> b(n, 0, false, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);

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_, k_] :=   b[n, 0, False, k]; Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 23 2017, translated from Maple *)

Formula

A(n,k) = Sum_{i=0..min(floor(n/2),k)} C(k,i) * i! * A258307(n,i).