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 (GCC 5.4.1) (judge verdict: Accepted)

#include <stdio.h>

int const M = 998244353;
int i = 1, n, x, c[7 << 14];

int pow(int a, int b) {
  long long r = 1;
  for(; b; b >>= 1) {
    if(b & 1)
      r = r * a % M;
    a = (long long)a * a % M;
  }
  return r;
}  

int main() {
  scanf("%d%d", &n, &x);
  int r = 0;
  if(x == 0) {
    for(c[x]++; i < n; c[x]++, i++) scanf("%d", &x);
    if(c[0] == 1) {
      r = 1;
      for(i = 1; c[i]; i++) {
        n -= c[i];
        r = (long long) r * pow(c[i - 1], c[i]) % M;
      }
    }
  }
  printf("%d\n", n > 1 ? 0 : r);
  return 0;
}