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 Python (3.4.3) (judge verdict: Wrong Answer)

import sys
from functools import lru_cache

@lru_cache(maxsize=None)

def kaijou(n):
    if n <= 1:
        return 1
    return kaijou(n-1)*n

def H(n,r):
    return kaijou(n+r-1)/kaijou(r)/kaijou(n-1)

from collections import Counter
n = int(input())
D = list(map(int,input().split()))
if D[0] != 0 or D.count(0) != 1:
    print(0)
    sys.exit()
D = Counter(D)
L = sorted(D.items())
pk = 0
pv = 1
ans = 1
for i,j in L:
    if i == 0:
        continue
    if i != pk+1:
        print(0)
        break
    ans *= pv**j
    ans %= 998244353
    pk = i
    pv = j
print(ans)