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.io.PrintWriter;
import java.util.Scanner;

public class Main {

    static Scanner sc = new Scanner(System.in);
    static PrintWriter out = new PrintWriter(System.out);

    public static void main(String[] args) {
        int n = nextInt();
        int[] d = new int[n];
        int[] dummy = new int[n];
        int max = 0;
        for(int i = 0; i < n; i++){
            d[i] = nextInt();
            if (max < d[i]) {
            	max = d[i];
            }
            dummy[d[i]] += 1;
        }

        boolean flag = true;
        int Answer = 0;

        if (d[0] != 0) {
        	flag = false;
        }
        for (int i = 1; i<n; i++) {
        	if (d[i] == 0) {
        		flag = false;
        	}
        	if (i <= max) {
        		if (dummy[i] == 0) {
        			flag = false;
        		}
        		Answer += dummy[i - 1 ]* dummy[i];
        	}
        	if (Answer >= 998244353) {
        		Answer = Answer % 998244353;
        	}
        }

        if (flag) {
        	println(Answer);
        } else {
        	println(0);
        }
    }

    static int nextInt() {
        return Integer.parseInt(next());
    }

    static String next() {
        return sc.next();
    }

    static String nextLine(){
        return sc.nextLine();
    }

    static void println(Object o) {
        out.println(o);
        out.flush();
    }

}