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).
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
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Jun 03 2011
STATUS
approved