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Problem 812

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Dynamical Polynomials

A dynamical polynomial is a monic polynomial $f(x)$ with integer coefficients such that $f(x)$ divides $f(x^2-2)$.

For example, $f(x) = x^2 - x - 2$ is a dynamical polynomial because $f(x^2-2) = x^4-5x^2+4 = (x^2 + x -2)f(x)$.

Let $S(n)$ be the number of dynamical polynomials of degree $n$.

For example, $S(2)=6$, as there are six dynamical polynomials of degree 2:
$$ x^2-4x+4 \quad x^2-x-2 \quad x^2-4 \quad x^2-1 \quad x^2+x-1 \quad x^2+2x+1 $$
Also, $S(5)=58$ and $S(20)=122087$.

Find $S(10\ 000)$. Give your answer modulo $998244353$.

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动态多项式

若首一整系数多项式$f(x)$整除$f(x^2-2)$，则称之为动态多项式。

例如，$f(x) = x^2 - x - 2$是动态多项式，因为$f(x^2-2) = x^4-5x^2+4 = (x^2 + x -2)f(x)$。

记$S(n)$为最高次数为$n$的动态多项式的数目。

例如，$S(2)=6$，因为共有六个最高次数为$2$的动态多项式：
$$ x^2-4x+4 \quad x^2-x-2 \quad x^2-4 \quad x^2-1 \quad x^2+x-1 \quad x^2+2x+1 $$
已知$S(5)=58$，$S(20)=122087$。

求$S(10\ 000)$，并将你的答案对$998244353$取余。

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