Inversion Number Squared

Consider sequences made of $a_1$ copies of $1$, $a_2$ copies of $2$, $\dots$, and $a_n$ copies of $n$. Pick one of these sequences uniformly at random. Find the expected value of the square of its inversion count, modulo $998\,244\,353$.

The inversion count of a sequence $p_1, p_2, \dots, p_n$ is the number of pairs $(i,j)$ with $i<j$ and $p_i>p_j$.

The first line contains $n$, the maximum value in the sequence. ($2 \leq n \leq 200\,000$)

The second line contains $a_i$, the number of copies of each value $i$, separated by spaces. ($1 \leq a_i$; $n \leq \sum_{i=1}^{n} a_i \leq 998\,244\,352$)

Print the answer modulo $998\,244\,353$.

The expected value of the square of the inversion count is a rational number. Write it as $\frac{p}{q}$ with $p$ and $q$ coprime. You can prove that $q$ is not a multiple of $998\,244\,353$. The answer is the smallest non-negative integer $a$ that satisfies $aq \equiv p \pmod{998\,244\,353}$.

There are two sequences made of one $1$ and one $2$: $12$ and $21$. Their inversion counts are $0$ and $1$, so the expected square is $\frac{1}{2}$. Since $499\,122\,177 \times 2 \equiv 1 \pmod{998\,244\,353}$, the answer is $499\,122\,177$.