KöMaL 2020 A776

Let$k>1$ be a fixed odd number, and for non-negative integers $n$ let $f_n=\sum_{\substack{0\le i\le n\\ k\mid n-2i}}\binom{n}{i}.$ Prove that $f_n$ satisfy the following recursion: $f_n^2=\sum_{i=0}^{n}\binom{n}{i}f_if_{n-i}.$