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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.BufferedReader;
import java.io.IOException;
import java.io.InputStream;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.HashMap;
import java.util.Map;

public class Main {
	private ModUtil modUtil = ModUtil.getInstance();

	public void exec() throws IOException {
		InputUtil input = new InputUtil();
		int n = input.nextInt();
		int[] a = input.nextInt(new int[n+1]);
		int[] search = new int[n+1];
		int l=0, r=0;
		Arrays.fill(search, 0);
		for (int i=0; i<a.length; i++) {
			if(search[a[i]-1] != 0) {
				l = search[a[i]-1];
				r = i+1;
				break;
			} 
			else search[a[i]-1] = i+1;
		}
		PrintWriter out = new PrintWriter(System.out);
		for (int k=1; k<=n+1; k++) {
			int full = modUtil.getC(n+1, k);
			int duplicate = 0;
			if (l-1+n+1-r >= k-1) {
				duplicate = modUtil.getC(l-1+n+1-r,k-1);
			}
			out.println(modUtil.sub(full,duplicate));
			
		}
		out.flush();
		input.close();
	}

	public static void main(String[] args) throws IOException {
		new Main().exec();
	}
}

class ModUtil {
	//Static Fields
	public static final int BASE_DEF = 1000000007;
	public static final int CAPACITY_F = 100001;
	private static Map<Integer,ModUtil> instance = new HashMap<Integer,ModUtil>();
	
	//Static Methods
	public static ModUtil getInstance() {return ModUtil.getInstance(BASE_DEF, CAPACITY_F);}
	public static ModUtil getInstance(int mod, int cap) {
		if(instance.containsKey(mod)){ 
			return instance.get(mod);
		} else {
			ModUtil mi = new ModUtil(mod,cap);
			instance.put(mod, mi);
			return mi;
		}
	}

	//Fields
	protected int mod = BASE_DEF;
	protected int capacity = CAPACITY_F;
	private int[] Farray = null;;
	private int[] FInvarray = null;

	//Constructors
	private ModUtil(int m, int cap) {
		mod = m;
		capacity=cap;
		initF();

	}
	
	//Methods
	//Initializer
	protected void initF() {
		if(Farray != null) return;
		Farray = new int[capacity+1];
		FInvarray = new int[capacity+1];
		for(int n=0; n<Farray.length; n++) {
			Farray[n] = (n==0) ? 1 : multiple(n, Farray[n-1]);
			FInvarray[n] = divide(1, Farray[n]);
		}
	}

	//Basic Opertion
	public int modulo(long v) {return (int)(v%mod);}
	public int add(int a, int b) {return modulo(a + b);}
	public int sub(int a, int b) {return add(a, mod - b);}
	public int multiple(int a, int b) {return modulo((long)a * modulo(b));}
	public int divide(int a, int b) {return multiple(a, power(b, mod-2));}
	
	public int divideF(int a, int fn) {
		if(fn<=capacity) {
			return multiple(a, getFInv(fn));
		}else{
			return divide(a, getF(fn));
		}
	}
	
	public int power(int a, int b) {
		int ret = 1;
		for (int bitMask = 1; bitMask<=b; bitMask<<=1) {
			if ( (b & bitMask) != 0 ) ret = multiple(ret, a);
			a = multiple(a, a);
		}
		return ret;
	}

	//Factorial Operation
	public int getF(int n) {return Farray[n];}
	public int getFInv(int n) {return FInvarray[n];}

	//Permutation Operation
	protected int calcP(int n, int r) {
		if (n < 0 || r < 0 || n < r) throw new IllegalArgumentException("n=" +n+ ",r=" +r);
		if (n <= 1 || r == 0) return 1;
		if (r == 1) return n;
		return divideF(getF(n), n-r);
		
	}

	public int getP(int n, int r) {return calcP(n, r);}

	//Combination Operation
	protected int calcC(int n, int r) {
		if(n < 0 || r < 0 || n < r) throw new IllegalArgumentException("n=" +n+ ",r=" +r);
		if(n == 0 || n == 1) return 1;
		if(n - r < r) r = n - r;
		return divideF(divideF(getF(n), r),n-r);
	}
	
	public int getC(int n, int r) {return calcC(n,r);}
	public int getH(int n, int r) {return calcC(n+r-1, r);}

}

class InputUtil {
	//Static Fields
	private static final int BUFSIZ = 600000;
	
	//Fields
	protected InputStream in;
	private byte[] buf = null;
	private int ptr = 0;
	private int max = -1;
	private BufferedReader br = null;
	
	//Constructors
	public InputUtil() {this(System.in,BUFSIZ);}
	public InputUtil(int bufsiz) {this(System.in,bufsiz);}
	public InputUtil(InputStream in) {this(in,BUFSIZ);}
	public InputUtil(InputStream in, int bufsiz) {
		initBuf(bufsiz);
		initIn(in);
	}
	
	//Methods
	//Initializer
	protected void initIn(InputStream in) {br = new BufferedReader(new InputStreamReader(this.in = in));}
	protected void initBuf(int bufsiz) {buf = new byte[bufsiz];}
	
	//String Input Reader/Parser
	private void readBuf() throws IOException {
		if(ptr >= max) {
			max = in.read(buf);
			ptr = 0;
		}
	}
	
	public BufferedReader getReader() {return br;}

	//Number Input Reader/Parser
	private byte now() {return buf[ptr];}
	private void next() throws IOException {ptr++;readBuf();}
	private boolean isNum(byte b) {return '0' <= b && b <= '9'; }
	private boolean isMinus(byte b) {return b=='-'; }
	
	public int nextInt() throws IOException {return nextInt(new int[1])[0];}
	public int[] nextInt(int[] result) throws IOException {
		readBuf();
		for(int i=0; i<result.length; i++) {
			int sign = 1;
			int res = 0;
			byte b;
			while (!isNum(b=now()) && !isMinus(b)) next();
			if (isMinus(now())) {
				sign = -1;
				next();
			}
			while (isNum(b=now())) {
				res *= 10;
				res += b - '0' ;
				next();
			}
			next();
			result[i] = sign * res;
		}
		return result;
	}
	
	//Close 
	public void close() throws IOException {
		if(!this.in.equals(System.in)) 
			this.in.close();
	}

}