102307: [AtCoder]ABC230 H - Bullion

Problem Statement

Takahashi won a claw machine competition and was awarded "all-you-can-stuff" gold blocks.
There are unlimited numbers of blocks weighing $w_1, w_2, \dots, w_K$ kilograms each, and an unlimited number of bags weighing $1$ kilogram each to stuff them.

Takahashi can bring home one non-empty bag.
A bag can contain zero or more other non-empty bags and zero or more gold blocks.

After arranging a truck with a load capacity of $W$ kilograms, he gets interested in the number of ways to stuff gold blocks and bring home a bag that weighs $w$ kilograms in total for $w = 2, 3, \dots, W$.
Find the number, modulo $998244353$, of possible states of the bag for each $w = 2, 3, \dots, W$. Here,

• two gold blocks are said to be the same when their weights are the same;
• two bags are said to be in the same state when the two multisets whose elements are the bags and gold blocks in the two bags are the same.

Constraints

• $2 \leq W \leq 2.5 \times 10^5$
• $1 \leq K \leq W$
• $1 \leq w_i \leq W$ $(1 \leq i \leq K)$
• $i \neq j \to w_i \neq w_j$ $(1 \leq i,j \leq K)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$W$ $K$
$w_1$ $w_2$ $\dots$ $w_K$

Output

Print the answer in $W - 1$ lines.
The $i$-th line should contain the count for $w = i + 1$.

Sample Input 1

4 1
1

Sample Output 1

1
2
4

The figure below enumerates the possible states of the bag for $w = 2, 3, 4$. (A circle represents a bag.)

Sample Input 2

10 10
1 2 3 4 5 6 7 8 9 10

Sample Output 2

1
3
7
18
45
121
325
904
2546

问题描述

高桥在抓娃娃机比赛中获胜，并获得了“随心所欲”黄金块的奖励。

有无限数量的每个重$w_1, w_2, \dots, w_K$千克的黄金块，以及无限数量的每个重$1$千克的袋子用来装它们。

高桥可以带回一个非空的袋子。

一个袋子可以包含零个或多个其他非空的袋子，以及零个或多个黄金块。

在安排了一个载重为$W$千克的卡车后，他对将黄金块装入袋子并带回家一个总重为$w$千克的袋子的方法数产生了兴趣，其中$w = 2, 3, \dots, W$。

对于每个$w = 2, 3, \dots, W$，找出袋子状态的可能性数，取模$998244353$。这里，

• 当两个黄金块的重量相同时，称这两个黄金块是相同的；
• 当两个袋子中包含的袋子和黄金块的多集相同时，称这两个袋子处于相同的状态。

约束条件

• $2 \leq W \leq 2.5 \times 10^5$
• $1 \leq K \leq W$
• $1 \leq w_i \leq W$ $(1 \leq i \leq K)$
• $i \neq j \to w_i \neq w_j$ $(1 \leq i,j \leq K)$
• 输入中的所有值都是整数。

输入

输入数据从标准输入以以下格式给出：

$W$ $K$
$w_1$ $w_2$ $\dots$ $w_K$

输出

在$W - 1$行中打印答案。

第$i$行应包含$w = i + 1$时的答案。

样例输入 1

4 1
1

样例输出 1

1
2
4

下图列出了当$w = 2, 3, 4$时袋子的可能状态。（圆圈表示一个袋子。）

样例输入 2

10 10
1 2 3 4 5 6 7 8 9 10

样例输出 2

1
3
7
18
45
121
325
904
2546