Bags with Balls
Description
There are $n$ bags, each bag contains $m$ balls with numbers from $1$ to $m$. For every $i \in [1, m]$, there is exactly one ball with number $i$ in each bag.
You have to take exactly one ball from each bag (all bags are different, so, for example, taking the ball $1$ from the first bag and the ball $2$ from the second bag is not the same as taking the ball $2$ from the first bag and the ball $1$ from the second bag). After that, you calculate the number of balls with odd numbers among the ones you have taken. Let the number of these balls be $F$.
Your task is to calculate the sum of $F^k$ over all possible ways to take $n$ balls, one from each bag.
The first line contains one integer $t$ ($1 \le t \le 5000$) — the number of test cases.
Each test case consists of one line containing three integers $n$, $m$ and $k$ ($1 \le n, m \le 998244352$; $1 \le k \le 2000$).
For each test case, print one integer — the sum of $F^k$ over all possible ways to take $n$ balls, one from each bag. Since it can be huge, print it modulo $998244353$.
Input
The first line contains one integer $t$ ($1 \le t \le 5000$) — the number of test cases.
Each test case consists of one line containing three integers $n$, $m$ and $k$ ($1 \le n, m \le 998244352$; $1 \le k \le 2000$).
Output
For each test case, print one integer — the sum of $F^k$ over all possible ways to take $n$ balls, one from each bag. Since it can be huge, print it modulo $998244353$.
Samples
<div class="test-example-line test-example-line-even test-example-line-0">5</div><div class="test-example-line test-example-line-odd test-example-line-1">2 3 8</div><div class="test-example-line test-example-line-even test-example-line-2">1 1 1</div><div class="test-example-line test-example-line-odd test-example-line-3">1 5 10</div><div class="test-example-line test-example-line-even test-example-line-4">3 7 2000</div><div class="test-example-line test-example-line-odd test-example-line-5">1337666 42424242 2000</div><div class="test-example-line test-example-line-odd test-example-line-5"></div>
1028
1
3
729229716
652219904