Problem:
Score : 300 points

Problem Statement

Given is an integer sequence D_1,...,D_N of N elements. Find the number, modulo 998244353, of trees with N vertices numbered 1 to N that satisfy the following condition:

For every integer i from 1 to N, the distance between Vertex 1 and Vertex i is D_i.

Notes

A tree of N vertices is a connected undirected graph with N vertices and N-1 edges, and the distance between two vertices are the number of edges in the shortest path between them.

Two trees are considered different if and only if there are two vertices x and y such that there is an edge between x and y in one of those trees and not in the other.

Constraints

1 \leq N \leq 10^5

0 \leq D_i \leq N-1

Input

Input is given from Standard Input in the following format:

N
D_1 D_2 ... D_N

Output

Print the answer.

Sample Input 1

4
0 1 1 2

Sample Output 1

2

For example, a tree with edges (1,2), (1,3), and (2,4) satisfies the condition.

Sample Input 2

4
1 1 1 1

Sample Output 2

0

Sample Input 3

7
0 3 2 1 2 2 1

Sample Output 3

24

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Linq;//リストの使用
using System.Collections.Generic;
using System.Text;//テキストの高速出力に必要
class Program
{
	static void Main()
	{
		int n = int.Parse(Console.ReadLine());
		long mod = 998244353;
		long answer = 1;
    long[] nums = Array.ConvertAll(Console.ReadLine().Split(' '),long.Parse);
    if(nums[0] != 0) answer = 0;
    Array.Sort(nums);

    int vertexMemo = 0;
    long lastlastCount = 1;
    long lastCount = 1;
    long countMemo = 0;

    for(int i = 1; i < n; i++)
    {
      if(nums[1] != 1) answer = 0;
    
      if(vertexMemo+1 == nums[i])
      {
        vertexMemo++;
        answer *= DivideMod(lastlastCount, lastCount, mod);
        answer %= mod;
        
        lastlastCount = lastCount;
        lastCount = 1;
        
      }else if(vertexMemo == nums[i])
      {
        lastCount++;
      }else answer = 0;

      //Console.WriteLine(lastlastCount+" "+lastCount+" "+answer);
    }

    answer *= DivideMod(lastlastCount, lastCount, mod);
    answer %= mod;

		Console.WriteLine(answer);
	}

  static long DivideMod(long x, long a, long p)//戻り値はx^a(mod p)。O(loga)
  {
    long num = 2;
    long answer = 1;
    long check = a;
    long memo = x%p;
    
    while(check > 0)
    {
      if(check % num == num / 2)
      {
        check -= num / 2;
        answer *= memo;
        answer %= p;
      }
    memo *= memo;
    memo %= p;
    num *= 2;
    }
    return answer;
  }
  
}