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 Haskell (GHC 7.10.3) (judge verdict: Time Limit Exceeded)

import Control.Applicative
import Data.List

m = 998244353
modm n = mod n m

mul' x y = modm $ modm x * modm y

power' x 0 = 1
power' x 1 = modm x
power' x n
  | (n `mod` 2) == 0 = let i = power' x (n `div` 2) in mul' i i
  | otherwise = mul' x $ power' x (n - 1)

div' x y = mul' x $ power' y (m - 2)

product' xs = foldl mul' 1 xs

factorial' n = product' [1..n]

npr' n r = product' [n - i | i <- [0..(r - 1)]]

ncr' n r = modm $ (npr' n r) `div'` (factorial' r)

main = do
  [n] <- map read <$> words <$> getLine :: IO [Int]
  xs <- map read <$> words <$> getLine :: IO [Int]
  --let n = 7
  --let xs = [0,3,2,1,2,2,1]
  let xs' = group $ sort xs
  let ls = map length xs'
  if (head xs /= 0) || ((length.head) xs' /= 1) then
    print 0
  else if and (zipWith (==) (sort $ nub xs) [0..]) then
    print $ foldl1' (mul') $ zipWith (power') (init ls) (tail ls)
  else
    print 0