Directable as Desired

AtCoder

IDarc155_f

Time5000ms

Memory256MB

Difficulty—

You are given a sequence of $N$ non-negative integers: $D=(D_1, D_2, \dots, D_N)$. Find the number of labeled trees with $N$ vertices numbered $1$ to $N$ that satisfy the following condition, modulo $998244353$. * There is a way to direct the $N-1$ edges so that the outdegree of each vertex $i\ (1\leq i \leq N)$ is $D_i$. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $0 \leq D_i \leq N-1$ * $\sum_{i=1}^{N} D_i = N-1$ * All values in the input are integers. ## Input The input is given from Standard Input in the following format: $N$ $D_1$ $D_2$ $\dots$ $D_N$ [samples]

Samples

Input #1

4
0 1 0 2

Output #1

5

Below are the five trees that satisfy the condition, directed in a desired way.
![image](https://img.atcoder.jp/arc155/5b5b99752b5330a2dd41607c3946fdd4.jpg)

Input #2

5
0 1 1 1 1

Output #2

Input #3

15
0 0 0 0 0 0 0 1 1 1 1 1 2 3 4

Output #3

63282877

API Response (JSON)

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    "name": "Directable as Desired",
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      "content": "You are given a sequence of $N$ non-negative integers: $D=(D_1, D_2, \\dots, D_N)$. Find the number of labeled trees with $N$ vertices numbered $1$ to $N$ that satisfy the following condition, modulo $",
      "description_type": "Markdown"
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    "platform": "AtCoder",
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      "content": "You are given a sequence of $N$ non-negative integers: $D=(D_1, D_2, \\dots, D_N)$.\nFind the number of labeled trees with $N$ vertices numbered $1$ to $N$ that satisfy the following condition, modulo $...",
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