Smallest Vertices

AtCoder

IDarc162_d

Time2000ms

Memory256MB

Difficulty—

In this problem, a rooted directed tree is a rooted tree where all edges are directed from the root to the leaves. You are given a sequence of non-negative integers $d=(d_1,d_2,\ldots,d_N)$ with a sum of $N-1$. Among the $N$\-vertex rooted directed trees with vertex numbered $1$ to $N$ and vertex $1$ as the root, a **good tree** is one that satisfies the following condition: * the out-degree of vertex $i\ (1\leq i \leq N)$ is $d_i$. Furthermore, for a vertex $v$ of a good tree, let $f(v)$ be the minimum vertex number of the vertices (including $v$) in the subtree rooted at vertex $v$, and $v$ is called a **good vertex** if it satisfies $f(v)=v$. Find the sum of the numbers of good vertices for all good trees, modulo $998244353$. ## Constraints * $2 \leq N \leq 500$ * $0 \leq d_i \leq N-1$ * $d_1 \geq 1$ * $\sum_{i=1}^N d_i = N-1$ * All input values are integers. ## Input The input is given from Standard Input in the following format: $N$ $d_1$ $d_2$ $\ldots$ $d_N$ [samples]

Samples

Input #1

4
2 0 1 0

Output #1

7

There are two good trees, as shown below. The blue vertices are good vertices.
![image](https://img.atcoder.jp/arc162/D-sample1-zFXKLnmt.png)
For these trees, there are $4$ and $3$ good vertices, respectively, so the answer is $7$.

Input #2

10
3 1 0 0 2 0 1 2 0 0

Output #2

API Response (JSON)

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    "name": "Smallest Vertices",
    "description": {
      "content": "In this problem, a rooted directed tree is a rooted tree where all edges are directed from the root to the leaves. You are given a sequence of non-negative integers $d=(d_1,d_2,\\ldots,d_N)$ with a sum",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc162_d"
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  "statements": [
    {
      "statement_type": "Markdown",
      "content": "In this problem, a rooted directed tree is a rooted tree where all edges are directed from the root to the leaves.\nYou are given a sequence of non-negative integers $d=(d_1,d_2,\\ldots,d_N)$ with a sum...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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