Problem Description

Little C has come up with a problem about bracket sequences, but he is not very good at creating test data, so he decided to use a random approach.

Little C specified the length \( N \) of the bracket sequence \( S \). Initially, \( S \) is filled with (.

He set an initial parameter \( K \) and follows the following random process until \( K = 0 \):

1. Uniformly randomly select an integer within the range \([1, N]\) and invert the bracket at that position in \( S \) (change ( to ) and vice versa).
2. If this operation reduces the number of (, decrement \( K \) by 1.

Now the data is created, and the problem is set.

Little C would like you to find the expected length of the longest valid subsequence (not necessarily contiguous) in \( S \) after the above operations, modulo \( 998244353 \).

Input Format

The first line of input contains two positive integers \( N \) and \( K \), as described in the problem statement.

Output Format

Output a single non-negative integer, which is the result of your calculation modulo \( 998244353 \).

Sample Input and Output

Sample Input #1

2 2

Sample Output #1

499122177

Sample Input #2

4 2

Sample Output #2

873463811

Sample Input #3

1919 810

Sample Output #3

488346634

Notes

Sample Explanation 1

The final bracket sequences can be )), (), or )( with probabilities \(\frac{1}{2}\), \(\frac{1}{4}\), and \(\frac{1}{4}\) respectively.

Their corresponding longest valid subsequence lengths are \(0, 2, 0\). Therefore, the final answer is \(\frac{1}{2}\), which is \(499122177\).

Data Range:

For the first \(5\%\) of the data, \(N = 1\);
Another \(5\%\) of the data, \(N = 2\);
Another \(5\%\) of the data, \(N \le 7\), \(K \le 5\);
Another \(15\%\) of the data, \(N \le 15\), \(K \le 500\);
Another \(15\%\) of the data, \(N \le 50\), \(K \le 50\);
Another \(15\%\) of the data, \(N \le 500\), \(K \le 100\);
For all data, \(1 \le N, K \le 5000\).

见题面 Input 部分

见题面 Output 部分