102817: [AtCoder]ABC281 Ex - Alchemy

Problem Statement

Takahashi has $A$ kinds of level-$1$ gems, and $10^{10^{100}}$ gems of each of those kinds.
For an integer $n$ greater than or equal to $2$, he can put $n$ gems that satisfy all of the following conditions into a cauldron to generate a level-$n$ gem in return.

• No two gems are of the same kind.
• Every gem's level is less than $n$.
• For every integer $x$ greater than or equal to $2$, there is at most one level-$x$ gem.

Find the number of kinds of level-$N$ gems that Takahashi can obtain, modulo $998244353$.

Here, two level-$2$ or higher gems are considered to be of the same kind if and only if they are generated from the same set of gems.

• Two sets of gems are distinguished if and only if there is a gem in one of those sets such that the other set does not contain a gem of the same kind.

Any level-$1$ gem and any level-$2$ or higher gem are of different kinds.

Constraints

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq A \leq 10^9$
• $N$ and $A$ are integers.

Input

The input is given from Standard Input in the following format:

$N$ $A$

Output

Print the answer.

Sample Input 1

3 3

Sample Output 1

10

Here are ten ways to obtain a level-$3$ gem.

• Put three kinds of level-$1$ gems into the cauldron.
  • Takahashi has three kinds of level-$1$ gems, so there is one way to choose three kinds of level-$1$ gems. Thus, he can obtain one kind of level-$3$ gem in this way.
• Put one kind of level-$2$ gem and two kinds of level-$1$ gems into the cauldron.
  • A level-$2$ gem can be obtained by putting two kinds of level-$1$ gems into the cauldron.
    • Takahashi has three kinds of level-$1$ gems, so there are three ways to choose two kinds of level-$1$ gems. Thus, three kinds of level-$2$ gems are available here.
  • There are three kinds of level-$2$ gems, and three ways to choose two kinds of level-$1$ gems, so he can obtain $3 \times 3 = 9$ kinds of level-$3$ gems in this way.

Sample Input 2

1 100

Sample Output 2

100

Sample Input 3

200000 1000000000

Sample Output 3

797585162

问题描述

高桥有$A$种等级为1的宝石，每种宝石有$10^{10^{100}}$个。

对于大于等于2的整数$n$，他可以将满足以下所有条件的$n$个宝石放入炼金锅中，以生成等级为$n$的宝石作为回报。

• 没有两个宝石是同一种类的。
• 每个宝石的等级都小于$n$。
• 对于大于等于2的整数$x$，最多有一个等级为$x$的宝石。

求高桥可以得到的等级为$N$的宝石种类数，对$998244353$取模。

这里，两个等级为2或更高的宝石被认为只有当它们由同一组宝石生成时才是同一种类。

• 只有当其中一个集合中存在一个宝石，另一个集合中没有同种类的宝石时，这两组宝石才能区分。

任何等级为1的宝石和任何等级为2或更高的宝石都是不同的种类。

约束条件

• $1 \leq N \leq 2 \times 10^5$
• $1 \leq A \leq 10^9$
• $N$和$A$都是整数。

输入

输入从标准输入中以以下格式给出：

$N$ $A$

输出

打印答案。

样例输入1

3 3

样例输出1

10

这里有十种方法可以得到一个等级为3的宝石。

• 将三种等级为1的宝石放入炼金锅中。
  • 高桥有三种等级为1的宝石，所以他有1种方法可以选择三种等级为1的宝石。因此，他可以通过这种方式得到一种等级为3的宝石。
• 将一种等级为2的宝石和两种等级为1的宝石放入炼金锅中。
  • 一个等级为2的宝石可以通过将两种等级为1的宝石放入炼金锅中得到。
    • 高桥有三种等级为1的宝石，所以他有3种方法可以选择两种等级为1的宝石。因此，这里有三种等级为2的宝石可用。
  • 这里有三种等级为2的宝石，以及三种选择两种等级为1的宝石的方法，所以他可以通过这种方式得到$3 \times 3 = 9$种等级为3的宝石。

样例输入2

1 100

样例输出2

100

样例输入3

200000 1000000000

样例输出3

797585162