题目描述
Problem Background
Amazing John had a dream where he was a girl from his past life.
She jumped into the ocean of OI, becoming a "Deep Sea Girl," maintaining the order of the ocean.
One day, the ocean was invaded by a swarm of fathead fish. To protect the deep sea, Amazing John and the experts fought the fish for 9 days and 9 nights, but the number of fish did not decrease.
Problem Description
After a long battle, Amazing John discovered a method to defeat the fathead fish:
There are $n$ "fathead fish" with "Divine Shield" and $m$ fathead fish without a shield. Each time, damage is inflicted with "Deadly Poison" on a living fathead fish with equal probability.
Now, Amazing John wants to know the expected number of damage infliction needed to kill all the fathead fish.
The answer should be modulo $998244353$.
"Divine Shield": When a fathead fish with a shield takes damage, it is immune to the damage. After immunity, the shield is broken.
"Fathead Fish": After a fathead fish's shield is broken, it grants a shield to all other non-dead, shieldless fathead fish.
"Deadly Poison": Instantly kills a fathead fish without a shield.
Input Format
The input consists of a single line containing two non-negative integers $n$ and $m$, representing the number of fathead fish with a shield and the number of fathead fish without a shield, respectively.
Output Format
The output consists of a single line with a non-negative integer $s$, representing the expected number of damage infliction modulo $998244353$.
Specifically, the answer can always be expressed in the form $\frac{p}{q}$ where $p, q \in \mathbb{N}$ and $q \neq 0$. You need to output the value of $p \times q^{-1}$ under modulo $998244353$.
Sample Input and Output
Sample Input #1
2 1
Sample Output #1
8
Sample Input #2
10 10
Sample Output #2
499122389
Sample Input #3
10 0
Sample Output #3
65
Sample Input #4
2 0
Sample Output #4
5
Notes
There are 20 data points, numbered from 1 to 20. For a subtask, you must pass all its data points to earn the score.
| Subtask | Data Points | Data Range | Score |
|---|---|---|---|
| 1 | 1-3 | $n, m \le 5 \times 10^3$ | 15 |
| 2 | 4-5 | $n \le 10^6$, $m = 0$ | 10 |
| 3 | 6-10 | $n, m \le 10^6$ | 25 |
| 4 | 11-14 | $n \le 10^{14}$, $m = 0$ | 20 |
| 5 | 15-20 | $n \le 10^{14}$, $m \le 10^6$ | 30 |
The form of the answer $\frac{p}{q}$ always satisfies $(p, q \in \mathbb{N}, 998244353 \nmid q)$.
I will secretly support you, don't tell anyone - Bob.
输入格式
见题面 Input 部分
输出格式
见题面 Output 部分