In our notes we were given the formula $$C(n)=\frac{1}{n+1}\binom{2n}{n}$$
It was proved by counting the number of paths above the line $y=0$ from $(0,0)$ to $(2n,0)$ using $n(1,1)$ up arrows and $n(1,-1)$ down arrows.
The notes are a bit unclear and I'm wondering if somebody could clarify for me. There is talk of reflections over the line $y=-1$ and how there are $\binom{2n}{n+1}$ paths from $(0,-2)$ to $(2n,0)$, but I can't see why this is. There is a similar proof on wikipedia but I'm more interested in this specific method.