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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified May 21 14:50 EDT 2019. Contains 323443 sequences. (Running on oeis4.)