ZS Shuffles Cards

题意翻译

有 $n+m$ 张牌，其中前 $n$ 张牌上分别标着 $1,2,\cdots,n$ 的数字，后 $m$ 张牌是鬼牌。现在我们打乱这些牌，然后开始抽牌游戏，每一轮你可以抽一张牌： - 如果抽到了一张标有数字 $x$ 的牌，就移除这张牌，并将 $x$ 加入一个集合 $S$ ； - 如果抽到了鬼牌，就把移除的牌重新加入牌堆，再次打乱所有牌的顺序，重新开始抽牌。如果你抽到了鬼牌，并且集合 $S$ 已经包括了 $[1,n]$ 中全部 $n$ 个数，那么抽牌游戏结束。 询问抽牌游戏结束的期望轮数。

题目描述

zscoder has a deck of $ n+m $ custom-made cards, which consists of $ n $ cards labelled from $ 1 $ to $ n $ and $ m $ jokers. Since zscoder is lonely, he wants to play a game with himself using those cards. Initially, the deck is shuffled uniformly randomly and placed on the table. zscoder has a set $ S $ which is initially empty. Every second, zscoder draws the top card from the deck. - If the card has a number $ x $ written on it, zscoder removes the card and adds $ x $ to the set $ S $ . - If the card drawn is a joker, zscoder places all the cards back into the deck and reshuffles (uniformly randomly) the $ n+m $ cards to form a new deck (hence the new deck now contains all cards from $ 1 $ to $ n $ and the $ m $ jokers). Then, if $ S $ currently contains all the elements from $ 1 $ to $ n $ , the game ends. Shuffling the deck doesn't take time at all. What is the expected number of seconds before the game ends? We can show that the answer can be written in the form $ \frac{P}{Q} $ where $ P, Q $ are relatively prime integers and $ Q \neq 0 \bmod 998244353 $ . Output the value of $ (P \cdot Q^{-1}) $ modulo $ 998244353 $ .

输入输出格式

输入格式

The only line of the input contains two integers $ n $ and $ m $ ( $ 1 \le n, m \le 2 \cdot 10^{6} $ ).

输出格式

Output a single integer, the value of $ (P \cdot Q^{-1}) $ modulo $ 998244353 $ .

输入输出样例

输入样例 #1

2 1

输出样例 #1

5

输入样例 #2

3 2

输出样例 #2

332748127

输入样例 #3

14 9

输出样例 #3

969862773

说明

For the first sample, it can be proven that the expected time before the game ends is $ 5 $ seconds. For the second sample, it can be proven that the expected time before the game ends is $ \frac{28}{3} $ seconds.