Eccoci.

A dynamical polynomial is a monicleading coefficient is $1$ 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 $2$:

$$ x^2-4x+4 \quad,\quad x^2-x-2 \quad,\quad x^2-4 \quad,\quad x^2-1 \quad,\quad x^2+x-1 \quad,\quad x^2+2x+1 $$

Also, $S(5)=58$ and $S(20)=122087$.

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