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++14 (GCC 5.4.1) (judge verdict: Accepted)

#include<bits/stdc++.h>

using namespace std;
typedef long long ll;
typedef string str;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
typedef vector<int> vi;
typedef vector<ll> vll;
typedef vector <pii> vpii;
typedef vector <pll> vpll;
typedef map <str,int> mapsi;
typedef map <str,int> :: iterator mapsitr;
typedef map <int , int> mint;
typedef map <ll , ll> mll;
typedef set <int> si;
typedef set <ll> sll;
typedef si :: iterator sitr;
typedef si :: reverse_iterator rsitr;
typedef sll :: iterator sltr;
typedef sll :: reverse_iterator rsltr;
#define mset multiset

typedef mset <int> msi;
typedef mset <ll> msll;
typedef msi :: iterator msitr;
typedef msi :: reverse_iterator msritr;
typedef msll :: iterator msltr;
typedef msll :: reverse_iterator mslritr;

#define mp make_pair
#define pb push_back
#define pob pop_back
#define pf push_front
#define pof pop_front
#define fi first
#define se second
#define fs first.second
#define ss second.second
#define ff first.first
#define sf second.first
#define newl '\n'

char to_upper (char x){
    if( 97 <= int(x) && int(x) <= 126)return char(x-32);
    else if( 65 <= int(x) && int(x) <= 90)return x;
}
char to_lower (char x){
    if( 97 <= int(x) && int(x) <= 126)return x;
    else if( 65 <= int(x) && int(x) <= 90)return char(x+32);
}
int numerize (char x){
    if(48 <= int(x) && int(x) <= 57)return int(x-'0');
    else if( 97 <= int(x) && int(x) <= 126)return int(x-96);
    else if( 65 <= int(x) && int(x) <= 90)return int(x-64);
}
bool isect (int l1, int r1, int l2, int r2){
    pii p1,p2;
    p1 = mp(l1,r1); p2 = mp(l2,r2);
    if(p1>p2)swap(p1,p2);
    if(p2.fi <= p1.se)return true;
    else return false;
}
ll fexp (ll num1, ll num2, ll MOD){
    if(num2==0)return 1;
    else if(num2==1)return num1;
    else{
        ll temp = fexp (num1,num2/2,MOD); ll res = ((temp%MOD) * (temp%MOD))%MOD;
        if(num2%2==1) res = ((res%MOD)*(num1%MOD))%MOD; return res;
    }
}
ll invmod (ll num, ll MOD){return fexp (num,MOD-2,MOD);}
ll gcd (ll num1, ll num2){
    if(num1 < num2) swap(num1,num2); ll num3 = num1 % num2 ;
    while(num3 > 0){ num1 = num2; num2 = num3; num3 = num1 % num2;}
    return num2;
}
ll lcm (ll num1 , ll num2){return (ll) (num1/gcd(num1,num2))*num2;}
// end of Template

ll n,z;
ll p[200000];
ll fac[200000];
ll md = 1e9+7;
ll calc (ll x, ll y){
    if(x<y)return 0;
    else{
        ll up,d1,d2,down;
        up = fac[x];
        d1 = fac[y];
        d2 = fac[x-y];
        down = (d1*d2)%md;
        down = invmod(down,md);
        ll res = (up * down) % md;
        return res;
    }
}
int main(){

    ios_base::sync_with_stdio(false);
    cin.tie(NULL);

    cin>>n;n++;
    for(int i=1;i<=n;i++){
        int temp; cin>>temp;
        if(!p[temp])p[temp]=i;
        else{z = (p[temp]-1)+(n-i);}
    }
    fac[0]=1;
    for(int i=1;i<=n;i++){
        fac[i] = ((fac[i-1]) * i)%md;
    }
    for(int k=1;k<=n;k++){
        ll ans = calc(n,k) - calc(z,k-1);
        if(ans<0)ans = md + ans;
        cout<<ans<<newl;
    }

}