A191390 Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n with k horizontal segments.

1, 0, 1, 1, 1, 0, 3, 2, 3, 1, 0, 8, 2, 5, 8, 7, 0, 22, 12, 1, 14, 22, 31, 3, 0, 64, 50, 12, 42, 64, 117, 28, 1, 0, 196, 184, 78, 4, 132, 196, 416, 162, 18, 0, 625, 648, 390, 52, 1, 429, 625, 1452, 762, 159, 5, 0, 2055, 2256, 1707, 392, 25, 1430, 2055, 5062, 3225, 1012, 85, 1, 0, 6917, 7868, 6954, 2280, 285, 6

Offset

0,7

Comments

A dispersed Dyck paths of length n is a Motzkin paths of length n with no (1,0)-steps at positive heights. A horizontal segment is a maximal sequence of consecutive (1,0)-steps.

Row n has 1 + ceiling(n/3) entries.

Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).

T(2n,0) = A000108(n) (the Catalan numbers); T(2n+1,0) = 0.

T(2n-1,1) = T(2n,1) = A014138(n) (partial sums of Catalan numbers).

Sum_{k>=0} k*T(n,k) = A191391(n).

Links

Table of n, a(n) for n=0..74.

Formula

G.f.: G(t,z) = (2*(1-z+t*z))/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)).

Example

T(5,2)=2 because we have (HH)UD(H) and (H)UD(HH), where U=(1,1), D=(1,-1), H=(1,0) (the horizontal segments are shown between parentheses).
Triangle starts:
  1;
  0,  1;
  1,  1;
  0,  3;
  2,  3,  1;
  0,  8,  2;
  5,  8,  7;
  0, 22, 12,  1;

Maple

G := (2*(1-z+t*z))/(1-z-t*z+(1-z+t*z)*sqrt(1-4*z^2)): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, k), k = 0 .. ceil((1/3)*n)) end do; # yields sequence in triangular form

Crossrefs

Cf. A000108, A001405, A014138, A191391.

Sequence in context: A324182 A166592 A103497 * A309698 A369997 A369605

Adjacent sequences: A191387 A191388 A191389 * A191391 A191392 A191393

Keyword

nonn,tabf

Author

Emeric Deutsch, Jun 03 2011

Status

approved