I noticed that when $n$ is an odd prime, the following congruence $$\binom{n-j-1}{k}+\binom{k+j}{k} \equiv 0 \pmod{n}$$ holds for $0 \le j \le \frac{(n-k)}2$ and odd values of $k$ such that $0 < k < n$.
Is there an identity that relates the above sum of binomials $\pmod{n}$ for any values of $n$ (no matter if it's prime or composite) and $k$ (odd and even values)?