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: Accepted)

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

const ll MOD = 998244353;
ll multip(ll x, ll y) { return (x*y)%MOD; }

int main() {
  int n; scanf("%d", &n);
  vector<int> a(n+1, 0);
  rep(i, n) {
    int x;
    scanf("%d", &x);
    if ((i==0)&&(x!=0)) a[n]++;
    a[x]++;
  }
  if ((a[0]!=1)||(a[n]==1)) {
    printf("0\n");
    return 0;
  }
  while (a[n]==0) n--;
  ll ans=1;
  rep(i, n) {
    rep(j, a[i+1]) ans=multip(ans, a[i]);
  }
  printf("%lld\n", ans);
  return 0;
}