A283595 Triangle read by rows: T(n,k) is the number of Motzkin prefixes (i.e., left factors of Motzkin paths) of length n and height k.
1, 1, 1, 1, 3, 1, 1, 7, 4, 1, 1, 15, 13, 5, 1, 1, 31, 38, 19, 6, 1, 1, 63, 105, 64, 26, 7, 1, 1, 127, 280, 202, 97, 34, 8, 1, 1, 255, 729, 612, 334, 139, 43, 9, 1, 1, 511, 1866, 1803, 1094, 516, 191, 53, 10, 1, 1, 1023, 4717, 5205, 3465, 1802, 760, 254, 64, 11, 1
Offset: 0
Examples
Triangle starts: 1; 1, 1; 1, 3, 1; 1, 7, 4, 1; 1, 15, 13, 5, 1; 1, 31, 38, 19, 6, 1; ... T(3,2) = 4 because we have UHU, HUU, UUD and UUH, where U=(1,1), D=(1,-1), H=(1,0). T(3,1) = 7 because we have UDH, HUD, UHD, UHH, HUH, HHU and UDU.
Links
- Alois P. Heinz, Rows n = 0..140, flattened
- Steven R. Finch, How far might we walk at random?, arXiv:1802.04615 [math.HO], 2018.
Programs
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Maple
b:= proc(x, y, m) option remember; `if`(x=0, z^m, b(x-1, y, m)+ `if`(y>0, b(x-1, y-1, m), 0)+b(x-1, y+1, max(m, y+1))) end: T:= n-> (p-> seq(coeff(p, z, i), i=0..n))(b(n, 0$2)): seq(T(n), n=0..12); # Alois P. Heinz, Mar 13 2017 -
Mathematica
b[x_, y_, m_] := b[x, y, m] = If[x==0, z^m, b[x-1, y, m] + If[y>0, b[x-1, y - 1, m], 0] + b[x-1, y+1, Max[m, y+1]]]; T[n_] := Function[p, Table[ Coefficient[p, z, i], {i, 0, n}]][b[n, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 18 2017, after Alois P. Heinz *)
Extensions
More terms from Alois P. Heinz, Mar 13 2017
Comments