Score : $475$ points

问题陈述

给定一个长度为 $N$ 的序列 $A = (A_1, A_2, \dots, A_N)$。对于每个 $k = 1, 2, \dots, N$，找出序列 $A$ 的长度为 $k$ 的（不一定是连续的）子序列的数量，这些子序列是等差数列。如果它们来自不同的位置，即使它们作为序列是相等的，也视为不同的子序列。

什么是子序列？序列 $A$ 的子序列是通过从 $A$ 中删除零个或多个元素，并在不改变剩余元素顺序的情况下排列得到的序列。

以上为大语言模型 kimi 翻译，仅供参考。

Problem Statement

You are given a sequence $A = (A_1, A_2, \dots, A_N)$ of length $N$. For each $k = 1, 2, \dots, N$, find the number, modulo $998244353$, of (not necessarily contiguous) subsequences of $A$ of length $k$ that are arithmetic sequences. Two subsequences are distinguished if they are taken from different positions, even if they are equal as sequences.

What is a subsequence? A subsequence of a sequence $A$ is a sequence obtained by deleting zero or more elements from $A$ and arranging the remaining elements without changing the order.

Constraints

• $1 \leq N \leq 80$
• $1 \leq A_i \leq 10^9$
• All input values are integers.

Input

The input is given from Standard Input in the following format:

$N$

$A_1$ $A_2$ $\dots$ $A_N$

Output

Print the answers for $k = 1, 2, \dots, N$ in this order, in a single line, separated by spaces.

Sample Input 1

5
1 2 3 2 3

Sample Output 1

5 10 3 0 0

• There are $5$ subsequences of length $1$, all of which are arithmetic sequences.
• There are $10$ subsequences of length $2$, all of which are arithmetic sequences.
• There are $3$ subsequences of length $3$ that are arithmetic sequences: $(A_1, A_2, A_3)$, $(A_1, A_2, A_5)$, and $(A_1, A_4, A_5)$.
• There are no arithmetic subsequences of length $4$ or more.

Sample Input 2

4
1 2 3 4

Sample Output 2

4 6 2 1

Sample Input 3

1
100

Sample Output 3

1