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)

# 2019/11/9

mod=998244353
n=int(input())
d=list(map(int,input().split()))

if d[0]!=0:
    print(0)
    exit()

res={}
for e in d:
    res[e]=res.get(e,0)+1


def cmb(n,r,mod=10**9+7):
    fac=[1,1]
    for i in range(2,n+1):
        fac.append(fac[i-1]*i%mod)
    ret=fac[n]*pow(fac[r]*fac[n-r]%mod,mod-2,mod)%mod
    return ret


ans=1
pre=res.get(1,-1)
if pre==-1:
    print(0)
    exit()

cnt=0
for i in range(2,len(res)):
    tmp=res.get(i,-1)
    if tmp==-1:
        print(0)
        exit()
    j=1
    while j<tmp:
        cnt+=cmb(tmp,j)
        j+=1
    cnt+=pre
    ans*=cnt
    cnt=0
    pre=tmp
print(ans)