Score : $300$ points

问题描述

有多少个长度为 $N$ 的整数序列 $A=(A_1, \ldots, A_N)$ 满足以下所有条件？

• $1\le A_i \le M$ $(1 \le i \le N)$

• $\displaystyle\sum _{i=1}^N A_i \leq K$

由于计数可能会非常庞大，因此请在模 $998244353$ 下求解。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

How many integer sequences of length $N$, $A=(A_1, \ldots, A_N)$, satisfy all of the conditions below?

• $1\le A_i \le M$ $(1 \le i \le N)$

• $\displaystyle\sum _{i=1}^N A_i \leq K$

Since the count can get enormous, find it modulo $998244353$.

Constraints

• $1 \leq N, M \leq 50$
• $N \leq K \leq NM$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$

Output

Print the answer.

Sample Input 1

2 3 4

Sample Output 1

6

The following six sequences satisfy the conditions.

• $(1,1)$
• $(1,2)$
• $(1,3)$
• $(2,1)$
• $(2,2)$
• $(3,1)$

Sample Input 2

31 41 592

Sample Output 2

798416518

Be sure to print the count modulo $998244353$.

update @ 2024/3/10 10:36:56