A258309 - OEIS

A258309

A(n,k) is the sum over all Motzkin paths of length n of products over all peaks p of (k*x_p+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, 4, 1, 1, 4, 7, 9, 1, 1, 5, 10, 23, 21, 1, 1, 6, 13, 43, 71, 51, 1, 1, 7, 16, 69, 151, 255, 127, 1, 1, 8, 19, 101, 261, 703, 911, 323, 1, 1, 9, 22, 139, 401, 1485, 2983, 3535, 835, 1, 1, 10, 25, 183, 571, 2691, 6973, 14977, 13903, 2188

(list; table; graph; refs; listen; history; text; internal format)

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! * A258310(n,i).

EXAMPLE

Square array A(n,k) begins:

: 1, 1, 1, 1, 1, 1, 1, ...

: 2, 3, 4, 5, 6, 7, 8, ...

: 4, 7, 10, 13, 16, 19, 22, ...

: 9, 23, 43, 69, 101, 139, 183, ...

: 21, 71, 151, 261, 401, 571, 771, ...

: 51, 255, 703, 1485, 2691, 4411, 6735, ...

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, 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, (k*x + 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, May 04 2017, translated from Maple *)

CROSSREFS

Columns k=0-1 give: A001006, A140456(n+2).

Main diagonal gives A261785.

Cf. A258306, A258310.

Sequence in context: A109225 A112564 A244911 * A197957 A089899 A092422

Adjacent sequences: A258306 A258307 A258308 * A258310 A258311 A258312

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, May 25 2015

STATUS

approved