Score : $600$ points

问题描述

你得到一个包含 $N$ 个非负整数的元组 $A=(A_1,A_2,\ldots,A_N)$，其中 $A_1=0$ 且 $A_N>0$。

高桥有 $N$ 个计数器。最初，所有计数器的值都是 $0$。 他将重复以下操作，直到对任意 $1\leq i\leq N$，第 $i$ 个计数器的值至少为 $A_i$。

均匀随机选择 $N$ 个计数器中的一个，并将其值设为 $0$。（每次选择都独立于其他计数器。） 将 其他 计数器的值增加 $1$。

计算并输出高桥重复该操作次数的期望值，结果对 $998244353$ 取模（参见注释）。

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

You are given a tuple of $N$ non-negative integers $A=(A_1,A_2,\ldots,A_N)$ such that $A_1=0$ and $A_N>0$.

Takahashi has $N$ counters. Initially, the values of all counters are $0$.
He will repeat the following operation until, for every $1\leq i\leq N$, the value of the $i$-th counter is at least $A_i$.

Choose one of the $N$ counters uniformly at random and set its value to $0$. (Each choice is independent of others.)
Increase the values of the other counters by $1$.

Print the expected value of the number of times Takahashi repeats the operation, modulo $998244353$ (see Notes).

Notes

It can be proved that the sought expected value is always finite and rational. Additionally, under the Constraints of this problem, when that value is represented as $\frac{P}{Q}$ using two coprime integers $P$ and $Q$, one can prove that there is a unique integer $R$ such that $R \times Q \equiv P\pmod{998244353}$ and $0 \leq R \lt 998244353$. Find this $R$.

Constraints

• $2\leq N\leq 2\times 10^5$
• $0=A_1\leq A_2\leq \cdots \leq A_N\leq 10^{18}$
• $A_N>0$
• All values in the input are integers.

Input

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

$N$

$A_1$ $A_2$ $\ldots$ $A_N$

Output

Print the expected value of the number of times Takahashi repeats the operation, modulo $998244353$.

Sample Input 1

2
0 2

Sample Output 1

6

Let $C_i$ denote the value of the $i$-th counter.

Here is one possible progression of the process.

• Set the value of the $1$-st counter to $0$, and then increase the value of the other counter by $1$. Now, $(C_1,C_2)=(0,1)$.
• Set the value of the $2$-nd counter to $0$, and then increase the value of the other counter by $1$. Now, $(C_1,C_2)=(1,0)$.
• Set the value of the $1$-st counter to $0$, and then increase the value of the other counter by $1$. Now, $(C_1,C_2)=(0,1)$.
• Set the value of the $1$-st counter to $0$, and then increase the value of the other counter by $1$. Now, $(C_1,C_2)=(0,2)$.

In this case, the operation is performed four times.

The probabilities that the process ends after exactly $1,2,3,4,5,\ldots$ operation(s) are $0,\frac{1}{4}, \frac{1}{8}, \frac{1}{8}, \frac{3}{32},\ldots$, respectively, so the sought expected value is $2\times\frac{1}{4}+3\times\frac{1}{8}+4\times\frac{1}{8}+5\times\frac{3}{32}+\dots=6$. Thus, $6$ should be printed.

Sample Input 2

5
0 1 3 10 1000000000000000000

Sample Output 2

874839568

update @ 2024/3/10 11:24:40