Tessellate S.T.E.M.S. (2019) - Mathematics - Category A (School) - Set 4 - Objective Problem 5

Fix a real number $a$ and a positive integer $n$ and consider the sequence $(x_k)_{k \geq 0}$ given by $x_0 = 0, x_1 = 1$ and $x_{k+2} = \frac{cx_{k+1} - (n-k)x_k}{k+1} \forall k \geq 0$ If $c$ be the largest real number for which $x_{n+1} = 0$ then for all $k \in \{ 1, \cdots , n \}$ , $x_k$ is given by

This problem is a part of Tessellate S.T.E.M.S. (2019)