ISL 2023 A6

Let $k \geqslant 2$ be an integer. Determine all sequences of positive integers $a_{1}, a_{2}, \ldots$ for which there exists a monic polynomial $P$ of degree $k$ with non-negative integer coefficients such that $$P\left(a_{n}\right)=a_{n+1} a_{n+2} \cdots a_{n+k}$$ for every integer $n \geqslant 1$. (Malaysia)