Closed formula for Catalan numbers

Statement

Let $n$ be a natural number, and let $C_n$ be the $n$-th Catalan number. Then $$C_n = \frac{1}{n + 1} \binom{2n}{n} = \frac{(2n)!}{(n + 1)!\, n!}.$$

Remarks

The fraction is an integer, because $C_n = \binom{2n}{n} - \binom{2n}{n+1}$. The numerator $\binom{2n}{n}$ is the central binomial coefficient. Example: $C_4 = \frac{1}{5} \binom{8}{4} = \frac{70}{5} = 14$.

Explanations

L1 human

Among the $\binom{2n}{n}$ grid paths from $(0, 0)$ to $(n, n)$, exactly one in $n + 1$ stays below the diagonal. So the Catalan number is the central binomial coefficient divided by $n + 1$.