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: Time Limit Exceeded)

using Yorukatsu055.Questions;
using Yorukatsu055.Extensions;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;

namespace Yorukatsu055.Questions
{
    public class QuestionF : AtCoderQuestionBase
    {
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var n = inputStream.ReadInt();
            var a = inputStream.ReadIntArray();

            var duplicated = a.GroupBy(i => i).Single(g => g.Count() > 1).Key;
            var leftIndex = -1;
            var rightIndex = -1;


            for (int i = 0; i < a.Length; i++)
            {
                if (a[i] == duplicated)
                {
                    if (leftIndex == -1)
                    {
                        leftIndex = i;
                    }
                    else
                    {
                        rightIndex = i;
                    }
                }
            }

            var left = leftIndex;
            var right = a.Length - rightIndex - 1;
            var center = rightIndex - leftIndex - 1;

            for (int length = 1; length <= a.Length; length++)
            {
                var total = Modular.Combination(a.Length, length);

                for (int l = Math.Max(0, length - right - 1); l <= left; l++)
                {
                    var r = length - l - 1;
                    if (r < 0)
                    {
                        break;
                    }

                    total -= Modular.Combination(left, l) * Modular.Combination(right, r);
                }

                yield return total.Value;
            }


        }

        [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.LayoutKind.Auto)]
        public struct Modular : IEquatable<Modular>, IComparable<Modular>
        {
            private const int _defaultMod = 1000000007;
            private readonly long _value;    // 0 <= value < Mod の値をとる
            public int Mod { get; }
            public int Value => (int)_value;

            public Modular(long value, int mod = _defaultMod)
            {
                if (mod < 2)
                {
                    throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
                }
                Mod = mod;

                if (value >= 0 && value < mod)
                {
                    _value = value;
                }
                else
                {
                    _value = value % mod;
                    if (Value < 0)
                    {
                        _value += mod;
                    }
                }
            }

            private static void CheckModEquals(Modular a, Modular b)
            {
                if (a.Mod != b.Mod)
                {
                    throw new ArgumentException($"{nameof(a)}, {nameof(b)}", $"両者の法（{nameof(Mod)}）は等しくなければなりません。");
                }
            }

            public static Modular operator +(Modular a, Modular b)
            {
                CheckModEquals(a, b);

                var result = a._value + b._value;
                if (result > a.Mod)
                {
                    result -= a.Mod;    // 剰余演算を避ける
                }
                return new Modular(result, a.Mod);
            }

            public static Modular operator -(Modular a, Modular b)
            {
                CheckModEquals(a, b);

                var result = a._value - b._value;
                if (result < 0)
                {
                    result += a.Mod;    // 剰余演算を避ける
                }
                return new Modular(result, a.Mod);
            }

            public static Modular operator *(Modular a, Modular b)
            {
                CheckModEquals(a, b);
                return new Modular(a._value * b._value, a.Mod);
            }

            public static Modular operator /(Modular a, Modular b)  // こいつは値渡しの方が速いっぽい
            {
                CheckModEquals(a, b);
                return a * Pow(b, a.Mod - 2);
            }

            // 需要は不明だけど一応
            public static bool operator ==(Modular left, Modular right) => left.Equals(right);
            public static bool operator !=(Modular left, Modular right) => !(left == right);
            public static bool operator <(Modular left, Modular right) => left.CompareTo(right) < 0;
            public static bool operator <=(Modular left, Modular right) => left.CompareTo(right) <= 0;
            public static bool operator >(Modular left, Modular right) => left.CompareTo(right) > 0;
            public static bool operator >=(Modular left, Modular right) => left.CompareTo(right) >= 0;

            public static explicit operator int(Modular a) => a.Value;
            public static explicit operator long(Modular a) => a._value;

            public static Modular Pow(Modular a, int n)
            {
                switch (n)
                {
                    case 0:
                        return new Modular(1, a.Mod);
                    case 1:
                        return a;
                    default:
                        var p = Pow(a, n / 2);
                        return p * p * Pow(a, n % 2);
                }
            }

            private static Dictionary<int, List<int>> FactorialCache { get; } = new Dictionary<int, List<int>>();
            private static Dictionary<int, int[]> FactorialInverseCache { get; } = new Dictionary<int, int[]>();
            private static Dictionary<int, List<int>> PermutationCache { get; } = new Dictionary<int, List<int>>();
            const int maxFactorial = 1000000;

            public static Modular Factorial(int n, int mod = _defaultMod)
            {
                if (n < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません。");
                }
                if (mod < 2)
                {
                    throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
                }

                if (!FactorialCache.ContainsKey(mod))
                {
                    FactorialCache.Add(mod, new List<int>() { 1 });
                }

                var cache = FactorialCache[mod];
                for (int i = cache.Count; i <= n; i++)  // Countが1（0!までキャッシュ済み）のとき1!～n!まで計算
                {
                    cache.Add((int)((long)cache[i - 1] * i % mod));
                }
                return new Modular(cache[n], mod);
            }

            public static Modular Permutation(int n, int r, int mod = _defaultMod)
            {
                CheckNR(n, r);

                if (!PermutationCache.ContainsKey(mod ^ n))
                {
                    PermutationCache.Add(mod ^ n, new List<int>() { 1 });
                }

                var cache = PermutationCache[mod ^ n];
                for (int i = cache.Count; i <= r; i++)  // Countが1（nP0までキャッシュ済み）のときnP1～nPrまで計算
                {
                    cache.Add((int)((long)cache[i - 1] * (n - (i - 1)) % mod));
                }
                return new Modular(cache[r], mod);
            }

            public static Modular Combination(int n, int r, int mod = _defaultMod)
            {
                if (!FactorialInverseCache.ContainsKey(mod))
                {
                    InitializeCombinationTable(maxFactorial, mod);
                }
                CheckNR(n, r);
                r = Math.Min(r, n - r);
                return n == r ? new Modular(1, mod) : new Modular(FactorialCache[mod][n], mod) * new Modular(FactorialInverseCache[mod][r], mod) * new Modular(FactorialInverseCache[mod][n - r], mod);
            }

            private static void InitializeCombinationTable(int max, int mod)
            {
                Factorial(max);
                FactorialInverseCache.Add(mod, new int[max + 1]);

                long fInv = (new Modular(1, mod) / Factorial(max, mod)).Value;
                FactorialInverseCache[mod][max] = (int)fInv;
                for (int i = max - 1; i >= 0; i--)
                {
                    fInv = (fInv * (i + 1)) % mod;
                    FactorialInverseCache[mod][i] = (int)fInv;
                }
            }

            public static Modular CombinationWithRepetition(int n, int r, int mod = _defaultMod) => Combination(n + r - 1, r, mod);

            private static void CheckNR(int n, int r)
            {
                if (n < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません");
                }
                if (r < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(r), $"{nameof(r)}は0以上の整数でなければなりません。");
                }
                if (n < r)
                {
                    throw new ArgumentOutOfRangeException($"{nameof(n)},{nameof(r)}", $"{nameof(r)}は{nameof(n)}以下でなければなりません。");
                }
            }

            public override string ToString() => $"{_value} (mod {Mod})";

            public bool Equals(Modular other) => _value == other._value && Mod == other.Mod;

            public int CompareTo(Modular other) => Mod == other.Mod ? _value.CompareTo(other._value) : 0;

            public override int GetHashCode() => Value ^ Mod;
        }

    }
}


namespace Yorukatsu055
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionF();    // 問題に合わせて書き換え
            var answers = question.Solve(Console.In);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
        }
    }
}

#region Base Classes

namespace Yorukatsu055.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);
    }

}

#endregion

#region Extensions

namespace Yorukatsu055.Extensions
{
    internal static class TextReaderExtensions
    {
        internal static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        internal static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        internal static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        internal static string ReadString(this TextReader reader) => reader.ReadLine();

        internal static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        internal static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        internal static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        internal static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);
    }
}

#endregion