Score : $500$ 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？

• $1\le A_i \le M$ $(1 \le i \le N)$

• $|A_i - A_{i+1}| \geq K$ $(1 \le i \le N - 1)$

由于满足条件的序列数量可能非常庞大，因此请在模 $998244353$ 下计算该数量。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

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

• $1\le A_i \le M$ $(1 \le i \le N)$

• $|A_i - A_{i+1}| \geq K$ $(1 \le i \le N - 1)$

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

Constraints

• $2 \leq N \leq 1000$
• $1 \leq M \leq 5000$
• $0 \leq K \leq M-1$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $M$ $K$

Output

Print the count modulo $998244353$.

Sample Input 1

2 3 1

Sample Output 1

6

The following $6$ sequences satisfy the conditions.

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

Sample Input 2

3 3 2

Sample Output 2

2

The following $2$ sequences satisfy the conditions.

• $(1,3,1)$
• $(3,1,3)$

Sample Input 3

100 1000 500

Sample Output 3

657064711

Print the count modulo $998244353$.

update @ 2024/3/10 10:47:45