AT_arc228_e [ARC228E] Pair of Permutations

Find the number, modulo $ 998244353 $ , of pairs of permutations $ (P,Q)=((P_1,P_2,\dots,P_N),(Q_1,Q_2,\dots,Q_N)) $ of $ (1,2,\dots,N) $ satisfying the following condition. - $ P_i \le i+M $ , $ Q_i \le i+M $ , and $ P_i \neq Q_i $ all hold for every integer $ i $ satisfying $ 1 \le i \le N $ .

The input is given from Standard Input in the following format: > $ N $ $ M $

Output the answer.

### Sample Explanation 1 The pairs $ (P,Q) $ satisfying the condition are the following four pairs. - $ P = (1,2,3),Q = (2,3,1) $ - $ P = (1,3,2),Q = (2,1,3) $ - $ P = (2,3,1),Q = (1,2,3) $ - $ P = (2,1,3),Q = (1,3,2) $ ### Constraints - $ 2 \le N \le 10^9 $ - $ 1 \le M \le 125000 $ - All input values are integers.