Ballot number
= Ballot number
The number of walks with $a$ up-steps and $b$ down-steps, $a\ge b$, that never go below zero is $\binom{a+b}{b}-\binom{a+b}{b-1}$. 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> $C_n$.