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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.IO;
using System.Collections.Generic;

namespace ARC077D
{
    public class Program
    {
        void Solve(StreamScanner ss, StreamWriter sw)
        {
            //---------------------------------
            var comb = new Combination(100010);
            var N = ss.Next(int.Parse) + 1;
            var A = ss.Next(int.Parse, N);

            var cnt = new int[N];
            foreach (var a in A) cnt[a]++;
            var dup = Array.IndexOf(cnt, 2);
            var left = Array.IndexOf(A, dup);
            var right = Array.LastIndexOf(A, dup);
            var M = N + left - right - 1;

            for (var i = 1; i <= N; i++) sw.WriteLine(comb.NCK(N, i) - comb.NCK(M, i - 1));
            //---------------------------------
        }

        static void Main()
        {
            var ss = new StreamScanner(new StreamReader(Console.OpenStandardInput()));
            var sw = new StreamWriter(Console.OpenStandardOutput()) {AutoFlush = false};
            new Program().Solve(ss, sw);
            sw.Flush();
        }

        static readonly Func<string, string> String = s => s;
    }

    public struct ModInt
    {
        public static int Mod = 1000000007;

        readonly long value;
        public int Value { get { return (int)value; } }
        public ModInt Inverse { get { return GetInverse(); } }

        public ModInt(long value)
        {
            value %= Mod;
            this.value = value < 0 ? value + Mod : value;
        }

        ModInt GetInverse()
        {
            if (value == 0) throw new InvalidOperationException("value must NOT equal 0");
            return Pow(Mod - 2);
        }

        public ModInt Square()
        {
            return this * this;
        }

        public ModInt Pow(long exp)
        {
            if (exp == 0) return 1;
            if (exp % 2 == 0) return Pow(exp / 2).Square();
            return this * Pow(exp - 1);
        }

        public static ModInt Parse(string str)
        {
            return long.Parse(str);
        }

        public static implicit operator ModInt(long value)
        {
            return new ModInt(value);
        }

        public static ModInt operator +(ModInt left, ModInt right)
        {
            return left.value + right.value;
        }

        public static ModInt operator -(ModInt left, ModInt right)
        {
            return left.value - right.value;
        }

        public static ModInt operator *(ModInt left, ModInt right)
        {
            return left.value * right.value;
        }

        public static ModInt operator /(ModInt left, ModInt right)
        {
            return left * right.Inverse;
        }

        public override bool Equals(object obj)
        {
            if (!(obj is ModInt)) return false;
            return value == ((ModInt)obj).value;
        }

        public override int GetHashCode()
        {
            return value.GetHashCode();
        }

        public override string ToString()
        {
            return value.ToString();
        }
    }

    public class Combination
    {
        readonly ModInt[] facts;

        public Combination(int max)
        {
            facts = new ModInt[max + 1];
            facts[0] = 1;
            for (var i = 1; i <= max; i++) facts[i] = facts[i - 1] * i;
        }

        public ModInt Fact(int n)
        {
            return facts[n];
        }

        public ModInt NCK(int n, int k)
        {
            if (n < k) return 0;
            return facts[n] / facts[k] / facts[n - k];
        }

        public ModInt NPK(int n, int k)
        {
            if (n < k) return 0;
            return facts[n] / facts[n - k];
        }

        public ModInt NHK(int n, int k)
        {
            if (k == 0) return 1;
            if (n == 0) return 0;
            return facts[n + k - 1] / facts[k] / facts[n - 1];
        }
    }

    public class StreamScanner
    {
        static readonly char[] Sep = {' '};
        readonly Queue<string> buffer = new Queue<string>();
        readonly TextReader textReader;

        public StreamScanner(TextReader textReader)
        {
            this.textReader = textReader;
        }

        public T Next<T>(Func<string, T> parser)
        {
            if (buffer.Count != 0) return parser(buffer.Dequeue());
            var nextStrings = textReader.ReadLine().Split(Sep, StringSplitOptions.RemoveEmptyEntries);
            foreach (var nextString in nextStrings) buffer.Enqueue(nextString);
            return Next(parser);
        }

        public T[] Next<T>(Func<string, T> parser, int x)
        {
            var ret = new T[x];
            for (var i = 0; i < x; ++i) ret[i] = Next(parser);
            return ret;
        }

        public T[][] Next<T>(Func<string, T> parser, int x, int y)
        {
            var ret = new T[y][];
            for (var i = 0; i < y; ++i) ret[i] = Next(parser, x);
            return ret;
        }
    }
}