Question 135
Let $C_{k} = \frac{1}{k+1} \binom{2k}{k}$ 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 $p$.
$$
\left\{\sum_{k=1}^{p-1} C_{k} n^{k} \mid n \in \{1,2, \ldots, p-1\}\right\}
$$
are divisible by $p$.