Convolution

You are given two sequences $a\_0, a\_1, \ldots, a\_n$ and $b\_0, b\_1, \ldots, b\_n$. You want to compute a new sequence $c\_0, c\_1, \ldots, c\_n$ such that $c\_k = \left(\sum\_{i = 0}^k {k \choose i} a\_i b\_{k-i}\right) \bmod 2^{32}\text{.}$ 

Here, ${k \choose i} = \frac{k!}{i! (k - i)!}$ are binomial coefficients.

Output $c\_0, c\_1, \ldots, c\_n$.\

The first line contains an integer $n$ ($1 \leq n \leq 2 \cdot 10^5$).

The second line contains $n + 1$ integers $a\_0, a\_1, \ldots, a\_n$ ($0\leq a\_i < 2^{32}$).

The third line contains $n + 1$ integers $b\_0, b\_1, \ldots, b\_n$ ($0 \leq b\_i < 2^{32}$).

Output one line with $n + 1$ integers: $c\_0, c\_1, \ldots, c\_n$.