Maximum Diameter

AtCoder

IDabc290_f

Time2000ms

Memory256MB

Difficulty—

For a sequence $X=(X_1,X_2\ldots,X_N)$ of length $N$ consisting of positive integers, we define $f(X)$ as follows: * A tree with $N$ vertices is said to be good if and only if the degree of the $i$\-th $(1 \leq i \leq N)$ vertex is $X_i$. If a good tree exists, $f(X)$ is the maximum diameter of a good tree; if it doesn't, $f(X)=0$. Here, the distance between two vertices is the minimum number of edges that must be traversed to travel from one vertex to the other, and the diameter of a tree is the maximum distance between two vertices. Find the sum, modulo $998244353$, of $f(X)$ over all possible sequences $X$ of length $N$ consisting of positive integers. We can prove that the sum of $f(X)$ is a finite value. Given $T$ test cases, find the answer for each of them. ## Constraints * $1\leq T \leq 2\times 10^5$ * $2 \leq N \leq 10^6$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format, where $\mathrm{test}_i$ denotes the $i$\-th test case: $T$ $\mathrm{test}_1$ $\mathrm{test}_2$ $\vdots$ $\mathrm{test}_T$ Each test case is given in the following format: $N$ [samples]

Samples

Input #1

Output #1

1
6
110
8052
9758476
421903645
377386885
881422708
120024839
351256142

If $N=3$,
for example,

*   when $X=(1,1,1)$, there is no tree with three vertices whose degrees are $1,1$, and $1$, so $f(X)=0$.
*   When $X=(2,1,1)$, the only possible tree is illustrated below. The diameter of this tree is $2$, so $f(X)=2$.

  
![image](https://img.atcoder.jp/abc290/7b4cd8233d2ee3eb307023bebaebd906.jpg)  
  

For $X=(2,1,1),(1,2,1),(1,1,2)$, we have $f(X)=2$; for other $X$, we have $f(X)=0$. Thus, the answer is $6$.

API Response (JSON)

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    "name": "Maximum Diameter",
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      "content": "For a sequence $X=(X_1,X_2\\ldots,X_N)$ of length $N$ consisting of positive integers, we define $f(X)$ as follows: *   A tree with $N$ vertices is said to be good if and only if the degree of the $i$",
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      "statement_type": "Markdown",
      "content": "For a sequence $X=(X_1,X_2\\ldots,X_N)$ of length $N$ consisting of positive integers, we define $f(X)$ as follows:\n\n*   A tree with $N$ vertices is said to be good if and only if the degree of the $i$...",
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