AT_arc232_e [ARC232E] Doubling

You are given a sequence of non-negative integers of length $ N+1 $ , $ X=(X_1,X_2,\ldots,X_{N+1}) $ , where $ X_{N+1}=0 $ . You can perform the following operation. - Choose an integer $ i $ satisfying $ 1 \leq i \leq N $ and $ X_i \geq 1 $ , subtract $ 1 $ from $ X_i $ , and add $ 2 $ to $ X_{i+1} $ . You are given an integer $ K $ . For each $ k=1,2,\ldots,K $ , find the number, modulo $ 998244353 $ , of sequences that can result from performing the operation exactly $ k $ times on the given sequence.

The input is given from Standard Input in the following format. $ X_{N+1} $ is not included in the input. > $ N $ $ K $ $ X_1 $ $ X_2 $ $ \ldots $ $ X_N $

Output the answers for $ k=1,2,\cdots,K $ in this order, separated by spaces, on a single line.

### Sample Explanation 1 The sequences that can result from performing the operation once are $ (0,3,0) $ and $ (1,0,2) $ , so there are two of them. The only sequence that can result from performing the operation twice is $ (0,2,2) $ , so there is one. Multiple orders of operations yield this sequence, but it is counted as one sequence. ### Sample Explanation 2 The initial sequence is $ (1,0,0) $ . With each operation, the sequence changes to $ (0,2,0) $ , $ (0,1,2) $ , $ (0,0,4) $ . The operation cannot be performed four times. ### Constraints - $ 1 \leq N \leq 20 $ - $ 1 \leq K \leq 250000 $ - $ 0 \leq X_i \leq 250000 $ $ (1 \leq i \leq N) $ - $ X_{N+1}=0 $ - All input values are integers.