OFFSET
0,6
COMMENTS
The Dyck path starts in red. At any point at any height > 0, the path can take a horizontal step and its color will change to blue. The color remains blue until the first time the path visits the y=0 line, at which point it changes to red again.
FORMULA
G.f: (sqrt(-4*z^2 + 1) + 1)/(z*sqrt(-4*z^2 + 1) + sqrt(-4*z^2 + 1) - z + 1).
D-finite with recurrence: n*a(n) -n*a(n-1) +6*(-n+2)*a(n-2) +3*(n-4)*a(n-3) +8*(n-3)*a(n-4) +4*(n-3)*a(n-5)=0. - R. J. Mathar, Jan 27 2020
EXAMPLE
For n=0, the only path is the empty path, so a(0)=1.
For n=1 and n=2, it is impossible to construct such a path, so a(1)=a(2)=0.
PROG
(PARI) a(n) = my(z='z+O('z^(n+1))); Vec((sqrt(-4*z^2 + 1) + 1)/(z*sqrt(-4*z^2 + 1) + sqrt(-4*z^2 + 1) - z + 1))[n+1] \\ Jianing Song, Nov 21 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Igor Kleiner, Aug 25 2019
STATUS
approved