HMIC 2020 Problem 4

Let $C_{k}=\frac{1}{k+1}\binom{2 k}{k}$ denote the $k^{\text {th }}$ Catalan number and $p$ be an odd prime. Prove that exactly half of the numbers in the set

$$\left\{\sum_{k=1}^{p-1} C_{k} n^{k} \mid n \in\{1,2, \ldots, p-1\}\right\}$$

are divisible by $p$.