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A282869
Triangle read by rows: T(n,k) is the number of dispersed Dyck prefixes (i.e., left factors of Motzkin paths with no (1,0) steps at positive heights) of length n and height k.
3
1, 1, 1, 1, 2, 1, 1, 4, 2, 1, 1, 7, 5, 2, 1, 1, 12, 10, 6, 2, 1, 1, 20, 21, 12, 7, 2, 1, 1, 33, 41, 28, 14, 8, 2, 1, 1, 54, 81, 56, 36, 16, 9, 2, 1, 1, 88, 155, 120, 72, 45, 18, 10, 2, 1, 1, 143, 297, 239, 165, 90, 55, 20, 11, 2, 1, 1, 232, 560, 492, 330, 220, 110, 66, 22, 12, 2, 1, 1, 376, 1054, 974, 715, 440, 286, 132, 78, 24, 13, 2, 1
OFFSET
0,5
COMMENTS
Row n has n+1 entries.
LINKS
Steven R. Finch, How far might we walk at random?, arXiv:1802.04615 [math.HO], 2018.
FORMULA
T(n,1) = A000071(n+1), (Fibonacci numbers minus 1).
EXAMPLE
Triangle starts:
1;
1, 1;
1, 2, 1;
1, 4, 2, 1;
1, 7, 5, 2, 1;
1, 12, 10, 6, 2, 1;
1, 20, 21, 12, 7, 2, 1;
...
T(4,3) = 2 because we have UUUD and HUUU, where U=(1,1), D=(1,-1), H=(1,0).
T(4,2) = 5 because we have UUDD, UUDU, UDUU, HUUD and HHUU.
MAPLE
b:= proc(x, y, m) option remember;
`if`(x=0, z^m, `if`(y>0, b(x-1, y-1, m), 0)+
`if`(y=0, b(x-1, y, m), 0)+b(x-1, y+1, max(m, y+1)))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..16); # Alois P. Heinz, Mar 13 2017
MATHEMATICA
b[x_, y_, m_] := b[x, y, m] = If[x == 0, z^m, If[y > 0, b[x - 1, y - 1, m], 0] + If[y == 0, b[x - 1, y, m], 0] + b[x - 1, y + 1, Max[m, y + 1]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, 0, 0]];
Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, May 12 2017, after Alois P. Heinz *)
CROSSREFS
Row sums give A000079.
T(2n,n) gives A283799.
Sequence in context: A209438 A106396 A384893 * A140998 A048004 A114394
KEYWORD
nonn,tabl
AUTHOR
Steven Finch, Feb 23 2017
STATUS
approved