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

0,5

Comments

Row n has n+1 entries.

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.

Example

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.

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 *)

Crossrefs

Row sums give A005773(n+1).

T(2n,n) gives A283667.

Cf. A000225, A097862, A282869.

Sequence in context: A140068 A179745 A121300 * A128119 A158198 A158793

Adjacent sequences: A283592 A283593 A283594 * A283596 A283597 A283598

Keyword

nonn,tabl

Author

Steven Finch, Mar 13 2017

Extensions

More terms from Alois P. Heinz, Mar 13 2017

Status

approved