201335: [AtCoder]ARC133 F - Random Transition

Memory Limit：1024 MB Time Limit：5 S
Judge Style：Text Compare Creator：
Submit：0 Solved：0

Description

Score : $1100$ points

Problem Statement

Given is an integer $N$.

Snuke is going to do the procedure below.

  • Let $x$ be a variable and initialize it with a random integer between $0$ and $N$ (inclusive). For each $0 \leq i \leq N$, $x=i$ is chosen with probability $A_i/10^9$.
  • Repeat the following operation $K$ times.
    • With probability $x/N$, decrease the value of $x$ by $1$; with probability $1-x/N$, increase it by $1$. (Here, note that $x$ will still be between $0$ and $N$ after the operation.)

For each $i=0,1,\cdots,N$, find the probability, modulo $998244353$, that $x=i$ after the procedure.

Definition of probability modulo $998244353$

It can be proved that the sought probabilities are always rational numbers. Additionally, under the Constraints of this problem, when representing each of them as an irreducible fraction $\frac{P}{Q}$, it can be proved that $Q \neq 0 \pmod{998244353}$. Thus, there uniquely exists an integer $R$ such that $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353$. Answer with this $R$.

Constraints

  • $1 \leq N \leq 100000$
  • $0 \leq A_i$
  • $\sum_{0 \leq i \leq N}A_i = 10^9$
  • $1 \leq K \leq 10^9$
  • All values in input are integers.

Input

Input is given from Standard Input in the following format:

$N$ $K$
$A_0$ $A_1$ $\cdots$ $A_N$

Output

For each $i=0,1,\cdots,N$, print the probability, modulo $998244353$, that $x=i$ after the procedure.

Sample Input 1

2 1
0 1000000000 0

Sample Output 1

499122177 0 499122177

First, $x$ is always initialized with $x=1$. In the subsequent operation, the value of $x$ is decreased by $1$ with probability $1/2$ and increased by $1$ with probability $1/2$. Eventually, we will have $x=0,1,2$ with probabilities $1/2,0,1/2$, respectively.

Sample Input 2

4 2
200000000 200000000 200000000 200000000 200000000

Sample Output 2

723727156 598946612 349385524 598946612 723727156

Sample Input 3

10 100
21265166 263511538 35931763 26849698 108140810 134702248 36774526 147099145 58335759 4118743 163270604

Sample Output 3

505314898 24510700 872096939 107940764 808162829 831195162 314651262 535843032 665830283 627881537 696038713

Input

题意翻译

给定整数 $n, k$ 和一个序列 $a$。 Snuke 会进行如下的操作： - 随机地取一个 $[0, n]$ 内的整数 $x$。对每个 $0\le i \le n$ ，$x = i$ 的概率为 $a_i / 10^9$。 - 进行如下操作 $k$ 次： - 以 $x / n$ 的概率将 $x$ 减 $1$；以 $1 - x / n$ 的概率将 $x$ 加 $1$。注意 $x\in [0, n]$ 总是成立的。 对每个 $0\le m \le n$，求出所有操作执行后 $x = m$ 的概率，对 $998244353$ 取模。 $1\le n \le 10^5, \ 0\le a_i\le 10^9, \ \sum a_i = 10^9, \ 1\le k \le 10^9$。

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