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
Offset: 0
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, ... : 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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..140, flattened
- Wikipedia, Motzkin number
Crossrefs
Programs
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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 *)
Formula
A(n,k) = Sum_{i=0..min(floor(n/2),k)} C(k,i) * i! * A258310(n,i).