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

0,6

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Wikipedia, Motzkin number

Formula

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

Example

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, ...

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 *)

Crossrefs

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

Main diagonal gives A266386.

Cf. A258307, A258309.

Sequence in context: A153899 A068098 A135722 * A049513 A121207 A097084

Adjacent sequences: A258303 A258304 A258305 * A258307 A258308 A258309

Keyword

nonn,tabl

Author

Alois P. Heinz, May 25 2015

Status

approved