This is a newbie question (I have never had much to do with Catalan combinatorics), so be prepared for an easy answer.
Consider the directed graph whose vertices are pairs $\left(a,b\right)$ of integers, and whose arcs are $\left(a,b\right) \to \left(a,b+1\right)$ and $\left(a,b\right) \to \left(a+1,b\right)$ (for all $\left(a,b\right) \in \mathbb{Z}^2$). We denote this directed graph by $\mathbb{Z}^2$ and draw it on the standard coordinate plane. Directed paths in this graph are called lattice paths.
Let $k$ be a positive integer. A result of Barbier (see, for example, Theorem 2 in I. P. Goulden, Luis G. Serrano, Maintaining the Spirit of the Reflection Principle when the Boundary has Arbitrary Integer Slope, Journal of Combinatorial Theory, Series A 104 (2003), pp. 317--326) says that if $n$ and $m$ are two nonnegative integers satisfying $n \geq km$, then the number of lattice paths from $\left(0,0\right)$ to $\left(m,n\right)$ that never go below the line $y=kx$ (that is, that contain no vertex of the form $\left(a,b\right)$ with $b<ka$) is $$\dfrac{n-km+1}{n+1}\dbinom{m+n}{m} = \dbinom{m+n}{m}-k\dbinom{m+n}{m-1}.$$ It makes sense to call these numbers "Fuss ballot numbers", since they generalize both the Fuss-Catalan numbers (obtained for $n=km$) and the ballot numbers (obtained for $k=1$).
(This result can easily be proven by induction over $m+n$. The paper I've linked is about finding more interesting proofs.)
I am wondering about the following:
If $n$ and $m$ are two nonnegative integers satisfying $n \leq km$, then what is the number of lattice paths from $\left(0,0\right)$ to $\left(m,n\right)$ that never go above the line $y=kx$ ?
Suprisingly, this does not seem to be an equivalent problem: The symmetries observed in both mentioned particular cases ($n=km$ and $k=1$) do not generalize (actually, these symmetries are different, so this is probably less surprising than it seems at first sight).