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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. 6
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 (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! * 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-Fran├žois 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

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Last modified August 20 10:05 EDT 2019. Contains 326149 sequences. (Running on oeis4.)