Let $n$ be a nonnegative integer. It is well-known that the number of lattice paths from $(0, 0)$ to $(n, n)$ with steps $(0, 1)$ and $(1, 0)$ that are never rising above the line $y=x$ is given by the Catalan number $C_n$. There are lots of generalisations of Catalan numbers in the literature; however, I have a different approach to these numbers which I could not manage to find any known results about.
Question: Let $a$ and $b$ be nonnegative integers. What is the number of lattice paths from $(0, 0)$ to $(a, b)$ with steps $(0, 1)$ and $(1, 0)$ that are never rising above the line $bx-ay=0$?
Let $C(a, b)$ be the integer valued function that counts the number of such lattice paths. It is clear that $$ C(n,n) = C_n = \frac{1}{n+1}\binom{2n}{n} $$ for every nonnegative integer $n$. We also know that $C(a,b)=C(b,a)$, which can be obtained by setting up a bijection between the two sets of such paths by means of reflecting the paths across the line $y=x$, so we can take $a≤b$ without loss of generality.
In the case of $a \mid b$, there is a well-known solution to the problem that is discussed by Hilton and Pedersen, Sands and possibly many others:
$$ C(a,ma) = \frac{1}{ma+1} \binom{a(m+1)}{a}. $$
Besides, there is a powerful enumeration tool called The Cycle Lemma (see Dvoretzky and Motzkin) which can be used to prove the equations mentioned above. Furthermore, it is possible to solve the problem thanks to the Cycle Lemma when $a$ and $b$ are coprime integers:
$$ gcd(a,b)=1 \Rightarrow C(a,b) = \frac{1}{a+b} \binom{a+b}{a}. $$
Unfortunately, none of my findings could prove itself useful in finding a general solution to the problem. Are there any existing papers or known results that shed some light on this question? If not, how can we achieve an explicit formula of $C(a,b)$?
