Fall 2016, problem 31
Do there exist natural numbers $n,k$ with $1\le k\le n-2$ such that $$\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 $$
Do there exist natural numbers $n,k$ with $1\le k\le n-2$ such that $$\binom{n}{k}^2+\binom{n}{k+1}^2=\binom{n}{k+2}^4 $$