Score : $550$ points

问题描述

有 $N$ 名人员站成一排，第 $i$ 个人站在从前往数的第 $i$ 个位置。

重复以下操作直到队伍中只剩一个人：

• 以 $\frac{1}{2}$ 的概率移除队伍最前面的人，否则将其移动至队伍末尾。

对于每个人员 $i=1,2,\ldots,N$，求在模 $998244353$ 下，人员 $i$ 成为最后剩下的人的概率。（每次选择移除或不移除都是随机且独立的。）

如何寻找模 $998244353$ 下的概率

本问题中所求的概率可以证明总是有理数。此外，此问题的约束条件保证了如果所求概率表示为不可约分数 $\frac{y}{x}$，则 $x$ 不会被 $998244353$ 整除。

现在，存在一个唯一的整数 $z$，满足 $0 \leq z \leq 998244352$（包含两端点），使得 $xz \equiv y \pmod{998244353}$。请报告这个 $z$ 值。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

There are $N$ people standing in a line, with person $i$ standing at the $i$-th position from the front.

Repeat the following operation until there is only one person left in the line:

• Remove the person at the front of the line with a probability of $\frac{1}{2}$, otherwise move them to the end of the line.

For each person $i=1,2,\ldots,N$, find the probability that person $i$ is the last person remaining in the line, modulo $998244353$. (All choices to remove or not are random and independent.)

How to find probabilities modulo $998244353$

The probabilities sought in this problem can be proven to always be a rational number. Furthermore, the constraints of this problem guarantee that if the sought probability is expressed as an irreducible fraction $\frac{y}{x}$, then $x$ is not divisible by $998244353$.

Now, there is a unique integer $z$ between $0$ and $998244352$, inclusive, such that $xz \equiv y \pmod{998244353}$. Report this $z$.

Constraints

• $2\leq N\leq 3000$
• All input values are integers.

Input

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

$N$

Output

Print the answer for people $i=1,2,\ldots,N$, separated by spaces.

Sample Input 1

2

Sample Output 1

332748118 665496236

Person $1$ will be the last person remaining in the line with probability $\frac{1}{3}$.

Person $2$ will be the last person remaining in the line with probability $\frac{2}{3}$.

Sample Input 2

5

Sample Output 2

235530465 792768557 258531487 238597268 471060930

update @ 2024/3/10 01:20:26