Nam Hyeon-uk

Time limit2sMemory limit1024 MB

Summary

Given n and m (0 to 3), sum the inversion counts of all length-n permutations with exactly m increasing subsequences of length 3, mod 998244353.

Level: Hard8 of 10
Topics: Combinatorics, Math, Dynamic programming
Solved: No attempts yet

Problem

Nam Hyeon-uk could not solve this problem.

Among all permutations of length $n$ , count those with exactly $m$ increasing subsequences (increasing subsequence) of length 3, and print the count modulo 998,244,353.

But Nam Hyeon-uk could not solve this problem.

So instead of counting the sequences, he wants to compute the sum of the inversion counts (inversion) of those sequences.

Print the answer to the problem Nam Hyeon-uk changed, modulo 998,244,353.

Input

The first line contains two integers $n$ and $m$ . ( $1 \le n \le 100000$ , $0 \le m \le 3$ )

Output

Print the answer on the first line.

Examples5

Example 1
Input
```
2 0
```
Expected output
```
1
```
Example 2
Input
```
3 2
```
Expected output
```
0
```
Example 3
Input
```
4 2
```
Expected output
```
3
```
Example 4
Input
```
5 0
```
Expected output
```
290
```
Example 5
Input
```
10 3
```
Expected output
```
1867456
```