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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. 4
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, ...

:  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-Fran├žois 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

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Last modified August 22 02:44 EDT 2019. Contains 326169 sequences. (Running on oeis4.)