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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;
#define rep(i, n) for(int i = 0; i < (int)(n); i++)
typedef pair<int, int> P;

long long int n, d[200000], c[200000];

int main(){
  cin >> n;
  rep(i, n) d[i] = 0;
  long long int mx = 0;
  bool flag = false;
  rep(i, n){
    long long int tmp;
    cin >> tmp;
    d[tmp]++;
    mx = max(mx, tmp + 1);
    if(i == 0 && tmp == 0){
      flag = true;
    }
  }
  long long int sum;
  if(flag){
    sum = 1;
  }else{
    sum = 0;
  }
  rep(i, mx) cerr << d[i] <<" "; cerr << endl;
  for(int i = 1; i < mx; i++){
    long long int a = d[i - 1], b = d[i];
    long long int tmp = 1;
    rep(j, b){
      tmp *= a;
      tmp %= 998244353;
    }
    cerr << tmp << endl;
    sum *= tmp;
    sum %= 998244353;
  }
  cout << sum << endl;
  return 0;
}