The Child and Binary Tree
Description
Our child likes computer science very much, especially he likes binary trees.
Consider the sequence of n distinct positive integers: c1, c2, ..., cn. The child calls a vertex-weighted rooted binary tree good if and only if for every vertex v, the weight of v is in the set {c1, c2, ..., cn}. Also our child thinks that the weight of a vertex-weighted tree is the sum of all vertices' weights.
Given an integer m, can you for all s (1 ≤ s ≤ m) calculate the number of good vertex-weighted rooted binary trees with weight s? Please, check the samples for better understanding what trees are considered different.
We only want to know the answer modulo 998244353 (7 × 17 × 223 + 1, a prime number).
The first line contains two integers n, m (1 ≤ n ≤ 105; 1 ≤ m ≤ 105). The second line contains n space-separated pairwise distinct integers c1, c2, ..., cn. (1 ≤ ci ≤ 105).
Print m lines, each line containing a single integer. The i-th line must contain the number of good vertex-weighted rooted binary trees whose weight exactly equal to i. Print the answers modulo 998244353 (7 × 17 × 223 + 1, a prime number).
Input
The first line contains two integers n, m (1 ≤ n ≤ 105; 1 ≤ m ≤ 105). The second line contains n space-separated pairwise distinct integers c1, c2, ..., cn. (1 ≤ ci ≤ 105).
Output
Print m lines, each line containing a single integer. The i-th line must contain the number of good vertex-weighted rooted binary trees whose weight exactly equal to i. Print the answers modulo 998244353 (7 × 17 × 223 + 1, a prime number).
2 3<br>1 2<br>
3 10<br>9 4 3<br>
5 10<br>13 10 6 4 15<br>
1<br>3<br>9<br>
0<br>0<br>1<br>1<br>0<br>2<br>4<br>2<br>6<br>15<br>
0<br>0<br>0<br>1<br>0<br>1<br>0<br>2<br>0<br>5<br>
Note
In the first example, there are 9 good vertex-weighted rooted binary trees whose weight exactly equal to 3:
