AmShZ Farm
题意翻译
给定 $n,k$,问有多少对正整数序列 $a,b$,满足
* $a$ 的长度为 $n$,$b$ 的长度为 $k$。$a,b$ 中的每个数都在 $[1,n]$。
* 将 $a$ 的每个数加上一个非负整数后可以得到一个 $1\dots n$ 的排列。
* 所有的 $a_{b_i}(1\le i\le k)$ 相等。
答案对 $998244353$ 取模。
题目描述
To AmShZ, all arrays are equal, but some arrays are more-equal than others. Specifically, the arrays consisting of $ n $ elements from $ 1 $ to $ n $ that can be turned into permutations of numbers from $ 1 $ to $ n $ by adding a non-negative integer to each element.
Mashtali who wants to appear in every problem statement thinks that an array $ b $ consisting of $ k $ elements is compatible with a more-equal array $ a $ consisting of $ n $ elements if for each $ 1 \le i \le k $ we have $ 1 \le b_i \le n $ and also $ a_{b_1} = a_{b_2} = \ldots = a_{b_k} $ .
Find the number of pairs of arrays $ a $ and $ b $ such that $ a $ is a more-equal array consisting of $ n $ elements and $ b $ is an array compatible with $ a $ consisting of $ k $ elements modulo $ 998244353 $ .
Note that the elements of $ b $ are not necessarily distinct, same holds for $ a $ .
输入输出格式
输入格式
The first line of input contains two integers $ n $ and $ k $ $ (1 \le n \le 10^9 , 1 \le k \le 10^5) $ .
输出格式
Print a single integer — the answer to the problem modulo $ 998244353 $ .
输入输出样例
输入样例 #1
1 1
输出样例 #1
1
输入样例 #2
2 2
输出样例 #2
8
输入样例 #3
5 4
输出样例 #3
50400
输入样例 #4
20 100
输出样例 #4
807645526
输入样例 #5
10000000 10000
输出样例 #5
883232350
说明
There are eight possible pairs for the second example:
1. $ a = \{1, 1\}, b = \{1, 1\} $
2. $ a = \{1, 1\}, b = \{1, 2\} $
3. $ a = \{1, 1\}, b = \{2, 1\} $
4. $ a = \{1, 1\}, b = \{2, 2\} $
5. $ a = \{1, 2\}, b = \{1, 1\} $
6. $ a = \{1, 2\}, b = \{2, 2\} $
7. $ a = \{2, 1\}, b = \{1, 1\} $
8. $ a = \{2, 1\}, b = \{2, 2\} $