Problem:
Score : 600 points

Problem Statement

You are given an integer sequence of length n+1, a_1,a_2,...,a_{n+1}, which consists of the n integers 1,...,n.
It is known that each of the n integers 1,...,n appears at least once in this sequence.

For each integer k=1,...,n+1, find the number of the different subsequences (not necessarily contiguous) of the given sequence with length k, modulo 10^9+7.

Notes

If the contents of two subsequences are the same, they are not separately counted even if they originate from different positions in the original sequence.

A subsequence of a sequence a with length k is a sequence obtained by selecting k of the elements of a and arranging them without changing their relative order. For example, the sequences 1,3,5 and 1,2,3 are subsequences of 1,2,3,4,5, while 3,1,2 and 1,10,100 are not.

Constraints

1 \leq n \leq 10^5

1 \leq a_i \leq n

Each of the integers 1,...,n appears in the sequence.

n and a_i are integers.

Input

Input is given from Standard Input in the following format:

n
a_1 a_2 ... a_{n+1}

Output

Print n+1 lines.
The k-th line should contain the number of the different subsequences of the given sequence with length k, modulo 10^9+7.

Sample Input 1

3
1 2 1 3

Sample Output 1

3
5
4
1

There are three subsequences with length 1: 1 and 2 and 3.

There are five subsequences with length 2: 1,1 and 1,2 and 1,3 and 2,1 and 2,3.

There are four subsequences with length 3: 1,1,3 and 1,2,1 and 1,2,3 and 2,1,3.

There is one subsequence with length 4: 1,2,1,3.

Sample Input 2

1
1 1

Sample Output 2

1
1

There is one subsequence with length 1: 1.

There is one subsequence with length 2: 1,1.

Sample Input 3

32
29 19 7 10 26 32 27 4 11 20 2 8 16 23 5 14 6 12 17 22 18 30 28 24 15 1 25 3 13 21 19 31 9

Sample Output 3

32
525
5453
40919
237336
1107568
4272048
13884156
38567100
92561040
193536720
354817320
573166440
818809200
37158313
166803103
166803103
37158313
818809200
573166440
354817320
193536720
92561040
38567100
13884156
4272048
1107568
237336
40920
5456
528
33
1

Be sure to print the numbers modulo 10^9+7.

Solution in Kotlin (1.0.0) (judge verdict: Accepted)

import java.util.*
import kotlin.*

import java.io.InputStream
import java.lang.NumberFormatException
import java.lang.StringBuilder


fun main(args: Array<String>) {
    Program().solve()
}

class Program {
    fun solve() {
        val sc = FastScanner()
        val n = sc.nextInt()
        val a = Array(n+1) { sc.nextInt() }

        // どの整数も必ず1回以上出現
        // 一つだけ2回出現する 残りは1回出現
        // 2回出現する整数をtとする

        var t = -1
        // 左側のtのインデックス
        var lt = -1
        // 右側
        var rt = -1

        val index = Array(n + 1) { -1 }
        for (i in a.indices) {
            if (index[a[i]] != -1) {
                t = a[i]
                lt = index[t]
                rt = i
                break
            }
            index[a[i]] = i
        }

        val fact = Array(200001) { ModInt(1) }
        for (i in 1 until 200000) {
            fact[i] = fact[i - 1] * i
        }

        fun C(n: Int, m: Int): ModInt {
            return if (n >= m) fact[n] * (fact[m] * fact[n - m]).inverse() else ModInt(0)
        }

        // 選び方のパターン (n+1) C i
        // tと lt未満、rt超過を選ぶパターンを引く

        val cnt = lt + (n - rt)
        val sb = StringBuilder()
        for (i in 1..n + 1) {
            if (i == 1) {
                sb.appendln(n)
                continue
            }

            sb.appendln(C(n + 1, i) - C(cnt, i - 1))
        }

        print(sb.toString())
    }
}

class ModInt(_num: Number) : Number() {
    companion object {
        const val mod: Long = 1000000007L
    }

    private val num: Long = if (_num.toLong() < 0) (_num.toLong() % mod) + mod else _num.toLong() % mod
    operator fun plus(right: Number) = ModInt(num + (right.toLong() % mod))
    operator fun minus(right: Number) = ModInt(num - (right.toLong() % mod))
    operator fun times(right: Number) = ModInt(num * (right.toLong() % mod))

    // modが素数の時のみ
    operator fun div(right: Number) = this * ModInt(right).inverse()

    fun power(n: Number): ModInt {
        var a = this
        var result = ModInt(1)
        var tmp = n.toLong()
        while (tmp > 0) {
            if (tmp % 2 == 1L) {
                result *= a
            }
            tmp /= 2
            a *= a
        }
        return result
    }

    // modが素数のときのみ 逆元
    fun inverse(): ModInt {
        return power(mod - 2L)
    }

    override fun toString() = num.toString()
    override fun toInt() = num.toInt()
    override fun toLong() = num
    override fun toByte() = num.toByte()
    override fun toChar() = num.toChar()
    override fun toDouble() = num.toDouble()
    override fun toFloat() = num.toFloat()
    override fun toShort() = num.toShort()
}

class FastScanner {
    companion object {
        val input: InputStream = System.`in`
        val buffer = ByteArray(1024) { 0 }
        fun isPrintableChar(c: Int): Boolean = c in 33..126
    }

    var ptr = 0
    var buflen = 0
    private fun hasNextByte(): Boolean {
        if (ptr < buflen) {
            return true
        } else {
            ptr = 0
            buflen = input.read(buffer)
            if (buflen <= 0) {
                return false
            }
        }
        return true
    }

    private fun readByte(): Int = if (hasNextByte()) buffer[ptr++].toInt() else -1

    private fun skipUnprintable() {
        while (hasNextByte() && !isPrintableChar(buffer[ptr].toInt())) ptr++
    }

    fun hasNext(): Boolean {
        skipUnprintable()
        return hasNextByte()
    }

    fun next(): String {
        if (!hasNext()) throw NoSuchElementException()
        val sb = StringBuilder()
        var b = readByte()
        while (isPrintableChar(b)) {
            sb.appendCodePoint(b)
            b = readByte()
        }
        return sb.toString()
    }

    fun nextInt(): Int {
        if (!hasNext()) throw NoSuchElementException()
        var n = 0
        var b = readByte()
        // '-' = 45
        val minus = b == 45
        if (minus) {
            b = readByte()
        }

        // '0' = 48 '9' = 57
        if (b !in 48..57) {
            throw NumberFormatException()
        }

        while (true) {
            if (b in 48..57) {
                n *= 10
                n += b - 48
            } else if (b == -1 || !isPrintableChar(b)) {
                return if (minus) -n else n
            } else {
                throw NumberFormatException()
            }
            b = readByte()
        }
    }

    fun nextLong(): Long {
        if (!hasNext()) throw NoSuchElementException()
        var n = 0L
        var b = readByte()
        // '-' = 45
        val minus = b == 45
        if (minus) {
            b = readByte()
        }

        // '0' = 48 '9' = 57
        if (b !in 48..57) {
            throw NumberFormatException()
        }

        while (true) {
            if (b in 48..57) {
                n *= 10
                n += b - 48
            } else if (b == -1 || !isPrintableChar(b)) {
                return if (minus) -n else n
            } else {
                throw NumberFormatException()
            }
            b = readByte()
        }
    }

    fun nextDouble(): Double = next().toDouble()
}