The Child and Binary Tree

我们的小朋友很喜欢计算机科学，而且尤其喜欢二叉树。 考虑一个含有 $n$ 个互异正整数的序列 $c_1,c_2\cdots,c_n$。如果一棵带点权的有根二叉树满足其所有顶点的权值都在集合 $\{c_1,c_2,\cdots,c_n\}$ 中，我们的小朋友就会将其称作神犇的。 并且他认为，一棵带点权的树的权值，是其所有顶点权值的总和。 给出一个整数 $m$，你能对于任意的 $1\leq s\leq m$ 计算出权值为 $s$ 的神犇二叉树的个数吗？请参照样例以更好的理解什么样的两棵二叉树会被视为不同的。 我们只需要知道答案关于 $998244353$ 取模后的值。 输入第一行有 $2$ 个整数 $n,m$ $(1\leq n,m\leq 10^5)$。 第二行有 $n$ 个用空格隔开的互异的整数 $c_1,c_2\cdots,c_n$ $(1\le c_i\le10^5)$。 输出 $m$ 行，每行有一个整数。第 $i$ 行应当含有权值恰为 $i$ 的神犇二叉树的总数。请输出答案关于 $998244353$ 取模的结果。

Our child likes computer science very much, especially he likes binary trees. Consider the sequence of $ n $ distinct positive integers: $ c_{1},c_{2},...,c_{n} $ . 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 $ {c_{1},c_{2},...,c_{n}} $ . 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 $ ( $ 7×17×2^{23}+1 $ , a prime number).

The first line contains two integers $ n,m $ $ (1<=n<=10^{5}; 1<=m<=10^{5}) $ . The second line contains $ n $ space-separated pairwise distinct integers $ c_{1},c_{2},...,c_{n} $ . $ (1<=c_{i}<=10^{5}) $ .

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 $ ( $ 7×17×2^{23}+1 $ , a prime number).

2 3
1 2

1
3
9

3 10
9 4 3

0
0
1
1
0
2
4
2
6
15

5 10
13 10 6 4 15

0
0
0
1
0
1
0
2
0
5

In the first example, there are $ 9 $ good vertex-weighted rooted binary trees whose weight exactly equal to $ 3 $ :