 Ballot number

The number of walks with $a$ up-steps and $b$ down-steps, $a\ge b$, that never go below zero is $\left(\genfrac{}{}{0ex}{}{a+b}{b}\right)-\left(\genfrac{}{}{0ex}{}{a+b}{b-1}\right)$. Reflecting the initial segment through the first step below zero pairs forbidden walks with walks having $b-1$ down-steps. Setting $a=b=n$ gives the Catalan number $C_n$.