2002 CMO 第 2 题

Given the polynomial sequence $(p_{n}(x))$ satisfying $p_{1}(x)=x^{2}-1$ , $p_{2}(x)=2x(x^{2}-1)$ , and $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$ , for $n\geq 2$ , let $s_{n}$ be the sum of the absolute values of the coefficients of $p_{n}(x)$ . For each $n$ , find a non-negative integer $k_{n}$ such that $2^{-k_{n}}s_n$ is odd.

给定多项式序列 $(p_{n}(x))$ 满足 $p_{1}(x)=x^{2}-1$ 、 $p_{2}(x)=2x(x^{2}-1)$ 和 $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$ ，对于$n\geq 2$ ，令 $s_{n}$ 为 $p_{n}(x)$ 的系数绝对值之和。对于每个 $n$ ，找到一个非负整数 $k_{n}$ ，使得 $2^{-k_{n}}s_n$ 为奇数。