P10461 [Template] Polynomial Composition of Set Power Series

Given a set power series $F(x)$ and a polynomial $G(x)$, it is guaranteed that $[x^{\varnothing}]F(x)=0$. Define the multiplication of $x$ as subset convolution. You need to compute, for every $S\subseteq\{1,2,\cdots,n\}$, the value of $[x^S]G(F(x))$ modulo $998244353$. If you still do not understand the statement, you can read the hint section at the end.

The first line contains a positive integer $n$. The next line contains $2^n$ non-negative integers. The $i$-th integer denotes $[x^S]F(x)$, where $a\in S$ if and only if, in the binary representation of $(i-1)$, the $a$-th bit (from low to high) is $1$. The next line contains $n+1$ non-negative integers. The $i$-th integer denotes $[x^{i-1}]G(x)$.

Output one line containing $2^n$ non-negative integers. The $i$-th integer denotes $[x^S]G(F(x))$ modulo $998244353$, where $a\in S$ if and only if, in the binary representation of $(i-1)$, the $a$-th bit (from low to high) is $1$.

**Constraints** For all testdata, it is guaranteed that $1\le n\le 20$, and $[x^S]F(x),[x^n]G(x)\in[0,998244353)\cap\mathbb Z$. This problem has $20$ test points. For the $i$-th test point, $n=i$. **Hint** Suppose $F(x)=\displaystyle \sum_S f_Sx^S$, then $[x^S]F(x)=f_S$. In this problem, the multiplication of $x$ is defined as subset convolution, i.e.: $$x^S\cdot x^T=\begin{cases}0&S\cap T\neq\varnothing\\x^{S\cup T}&\text{otherwise}\end{cases}$$ **Please pay attention to the efficiency differences caused by contiguous memory access.** Translated by ChatGPT 5