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E - Distance Sequence

Score : $500$ points

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 $998244353$.

2 3 1

6

The following $6$ sequences satisfy the conditions.

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

3 3 2

2

The following $2$ sequences satisfy the conditions.

• $(1,3,1)$
• $(3,1,3)$

100 1000 500

657064711

Print the count modulo $998244353$.