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 Java8 (OpenJDK 1.8.0) (judge verdict: Wrong Answer)

import java.util.*;

public class Main {
  
  public static void main(String[] args) {
    Scanner sc = new Scanner(System.in);
    int N = Integer.parseInt(sc.nextLine());
    int[] D = new int[N];
    String next = sc.nextLine();
    String[] Ds = next.split(" ");
    for(int i = 0; i < N; i++) {     
      D[i] = 0;
    }
    //1は距離ゼロでなければならない
    if(!Ds[0].equals("0")) {
      System.out.println(0);
      return;
    }
    for(int i = 0; i < N; i++) {
      int a = Integer.parseInt(Ds[i]);
      if(a == N) { //Nにはたどり着けない
        System.out.println(0);
        return;
      }
      D[a] ++;
    }
    //D[0]は必ず1でなければならない
    if(D[0] != 1) {
      System.out.println(0);
      return;
    }
    int sum = 1;
    boolean flag = false;
    for(int i = N - 1; i >= 1; i--) {   
      if(D[i] == 0 && flag) {
        sum = 0;
      }
      if(D[i] != 0 && !flag) {
        flag = true;
      }
      sum *= (int) Math.pow(D[i - 1], D[i]);
    }
    System.out.println(sum);
  }
}