Call a Dyck path even when its height is even and call it odd if its height is odd. Let $E_n$ be the number of even Dyck paths from $(0,0)$ to $(2n-2,0)$ and $O_n$ the odd ones.
Question 1: For which $n$ is the sequence $a_n=E_n-O_n$ positive? Is it infinitely often negative and infinitely often positive? Can $a_n$ be zero for $n>3$?
Question 2: Is there a good upper bound for $|a_n|$?
