1997 Putnam A6

For a positive integer $n$ and any real number $c$, define

$x_k$ recursively by $x_0=0$, $x_1=1$, and for $k\geq 0$,

$$x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}.$$

Fix $n$ and then take $c$ to be the largest value for which $x_{n+1}=0$.

Find $x_k$ in terms of $n$ and $k$, $1\leq k\leq n$.

对于正整数 $n$ 和任何实数 $c$，定义

$x_k$ 通过 $x_0=0$、$x_1=1$ 递归，并且对于 $k\geq 0$，

$$x_{k+2}=\frac{cx_{k+1}-(n-k)x_k}{k+1}。$$

固定$n$，然后取$c$为$x_{n+1}=0$时的最大值。

根据 $n$ 和 $k$、$1\leq k\leq n$ 求 $x_k$。