CF1349D.Slime and Biscuits

Slime and his $n$ friends are at a party. Slime has designed a game for his friends to play.

At the beginning of the game, the $i$-th player has $a_i$ biscuits. At each second, Slime will choose a biscuit randomly uniformly among all $a_1 + a_2 + \ldots + a_n$ biscuits, and the owner of this biscuit will give it to a random uniform player among $n-1$ players except himself. The game stops when one person will have all the biscuits.

As the host of the party, Slime wants to know the expected value of the time that the game will last, to hold the next activity on time.

For convenience, as the answer can be represented as a rational number $\frac{p}{q}$ for coprime $p$ and $q$, you need to find the value of $(p \cdot q^{-1})\mod 998\,244\,353$. You can prove that $q\mod 998\,244\,353 \neq 0$.

史莱姆和他的 $n$ 个朋友正在参加一场派对。史莱姆为朋友们设计了一个游戏。

游戏开始时，第 $i$ 位玩家拥有 $a_i$ 块饼干。每一秒，史莱姆会从全部 $a_1 + a_2 + \ldots + a_n$ 块饼干中均匀随机地选出一块；该饼干当前的持有者将这块饼干均匀随机地送给其余 $n-1$ 位玩家中的一位（即不能送给自己）。当某一位玩家拥有了全部饼干时，游戏结束。

作为派对主持人，史莱姆希望知道游戏持续时间的期望值，以便准时安排下一项活动。

为方便起见，由于答案可表示为既约分数 $\frac{p}{q}$（其中 $p$ 与 $q$ 互质），你需要计算 $(p \cdot q^{-1}) \bmod 998\,244\,353$。可以证明 $q \bmod 998\,244\,353 \neq 0$。

The first line contains one integer $n\ (2\le n\le 100\,000)$: the number of people playing the game.

The second line contains $n$ non-negative integers $a_1,a_2,\dots,a_n\ (1\le a_1+a_2+\dots+a_n\le 300\,000)$, where $a_i$ represents the number of biscuits the $i$-th person own at the beginning.

第一行包含一个整数 $n\ (2\le n\le 100\,000)$：参与游戏的人数。

第二行包含 $n$ 个非负整数 $a_1,a_2,\dots,a_n\ (1\le a_1+a_2+\dots+a_n\le 300\,000)$，其中 $a_i$ 表示第 $i$ 个人初始拥有的饼干数量。

Print one integer: the expected value of the time that the game will last, modulo $998\,244\,353$.

输出一个整数：游戏持续时间的期望值对 $998\,244\,353$ 取模的结果。

For the first example, in the first second, the probability that player $1$ will give the player $2$ a biscuit is $\frac{1}{2}$, and the probability that player $2$ will give the player $1$ a biscuit is $\frac{1}{2}$. But anyway, the game will stop after exactly $1$ second because only one player will occupy all biscuits after $1$ second, so the answer is $1$.

对于第一个样例，在第一秒内，玩家 $1$ 将饼干给玩家 $2$ 的概率为 $\frac{1}{2}$，而玩家 $2$ 将饼干给玩家 $1$ 的概率也为 $\frac{1}{2}$。但无论如何，游戏将在恰好 $1$ 秒后结束，因为经过 $1$ 秒后，将只有一名玩家持有全部饼干，因此答案为 $1$。