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 Go (1.6) (judge verdict: Wrong Answer)

package main

import (
	"bufio"
	"fmt"
	"os"
	"strconv"
)

func out(x ...interface{}) {
	fmt.Println(x...)
}

var sc = bufio.NewScanner(os.Stdin)

func getInt() int {
	sc.Scan()
	i, e := strconv.Atoi(sc.Text())
	if e != nil {
		panic(e)
	}
	return i
}

func getString() string {
	sc.Scan()
	return sc.Text()
}

// min, max, asub, absなど基本関数
func max(a, b int) int {
	if a > b {
		return a
	}
	return b
}

func min(a, b int) int {
	if a < b {
		return a
	}
	return b
}

func asub(a, b int) int {
	if a > b {
		return a - b
	}
	return b - a
}

func abs(a int) int {
	if a >= 0 {
		return a
	}
	return -a
}

const mod = 998244353

func mpow(a, b int) int {
	if b == 0 {
		return 1
	} else if b%2 == 0 {
		tmp := mpow(a, b/2)
		return tmp * tmp % mod
	}
	return mpow(a, b-1) * a % mod
}

var fracMemo = []int{1, 1}

func mfrac(n int) int {
	if len(fracMemo) > n {
		return fracMemo[n]
	}
	if len(fracMemo) == 0 {
		fracMemo = append(fracMemo, 1)
	}
	for len(fracMemo) <= n {
		size := len(fracMemo)
		fracMemo = append(fracMemo, fracMemo[size-1]*size%mod)
	}
	return fracMemo[n]
}

var ifracMemo = []int{1, 1}

func mifrac(n int) int {
	if len(ifracMemo) > n {
		return ifracMemo[n]
	}
	if len(ifracMemo) == 0 {
		fracMemo = append(ifracMemo, 1)
	}
	for len(ifracMemo) <= n {
		size := len(ifracMemo)
		ifracMemo = append(ifracMemo, ifracMemo[size-1]*mpow(size, mod-2)%mod)
	}
	return ifracMemo[n]
}

func nCr(n, r int) int {
	if n == r {
		return 1
	}
	if n < r || r < 0 {
		return 0
	}
	ret := 1
	ret *= mfrac(n)
	ret %= mod
	ret *= mifrac(r)
	ret %= mod
	ret *= mifrac(n - r)
	ret %= mod
	return (ret)
}

func nHr(n, r int) int {
	return nCr(n+r-1, r)
}

func main() {
	sc.Split(bufio.ScanWords)
	N := getInt()
	d := make(map[int]int)
	for i := 0; i < N; i++ {
		x := getInt()
		if i == 0 && x != 0 {
			out(0)
			return
		}
		d[x]++
	}

	ans := 1
	cnt := d[0] + d[1]
	for i := 2; i < N; i++ {
		y := d[i-1]
		x := d[i]
		if x == 0 {
			break
		}
		// out(y, x, mpow(y, x))
		ans *= mpow(y, x)
		ans %= mod

		cnt += x
		if cnt == N {
			break
		}
	}
	// out("N", cnt, N)
	if cnt != N {
		out(0)
		return
	}
	out(ans)
}