Question 185
Determine, with proof, all positive integers $k$ such that
$$
\frac{1}{n+1} \sum_{i=0}^{n} \binom{n}{i}^{k}
$$
is an integer for every positive integer $n$.
$$
\frac{1}{n+1} \sum_{i=0}^{n} \binom{n}{i}^{k}
$$
is an integer for every positive integer $n$.