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 Haskell (GHC 7.10.3) (judge verdict: Accepted)

import qualified Data.ByteString.Char8 as BC
import qualified Data.IntMap as M
import Data.Maybe (fromJust)
import Data.List
import Data.Array.IO
import Control.Monad

type Memo = IOUArray Int Int

modN :: Int
modN = 10^9+7

main = do
  n <- readLn :: IO Int
  a <- getIntListBC
  check <- newArray (1, n) 0 :: IO Memo
  l <- numbering check a 1
  table <- newArray (0, n+1) 0 :: IO Memo
  makeTable table (n+1) (n+1)
  let k = [1..n+1]
      dup = n - l - 1
      cn = fact (n+1)
      cnl = fact dup
  ans <- mapM (solve table cn cnl n l) k
  putStr $ unlines . map show $ ans

bsToInt :: BC.ByteString -> Int
bsToInt = fst . fromJust . BC.readInt

getIntListBC :: IO [Int]
getIntListBC = map bsToInt . BC.words <$> BC.getLine

numbering :: Memo -> [Int] -> Int -> IO Int
numbering memo [] n = return 0
numbering memo (i:is) n = do
  c <- readArray memo i
  case c of
    0 -> do
      writeArray memo i n
      return =<< numbering memo is $ n + 1
    _ -> do
      return $ n - c - 1

--x^n (mod modN)
(|^) :: Int -> Int -> Int
(|^) x n
  | n == 0    = 1
  | even n    = (mod (x * x) modN) |^ (n `div` 2)
  | otherwise = (flip mod modN) . (*x) $ x |^ (n - 1)

--x! (mod modN)
fact :: Int -> Int
fact n
  | n == 0    = 1
  | otherwise = (flip mod modN) . (*n) $ fact (n-1)

makeTable :: Memo -> Int -> Int -> IO Memo
makeTable table n (-1) = return table
makeTable table n i
  | n == i = do
    writeArray table i $ (fact n) |^ (10^9+5)
    makeTable table i $ i-1
  | otherwise = do
    ie <- readArray table (i+1)
    writeArray table i $ flip mod modN $ ie * (i+1)
    makeTable table i $ i-1

comb :: Memo -> Int -> Int -> Int -> IO Int
comb memo cn n k
  |n >= k = do
    ik <- readArray memo k
    ink <- readArray memo $ n - k
    let tmp = flip mod modN $ cn * ik
        ans = flip mod modN $ tmp * ink
    return ans
  | otherwise = return 0

solve :: Memo -> Int -> Int -> Int -> Int -> Int -> IO Int
solve table cn cnl n l k = do
  x <- comb table cn (n+1) k
  y <- comb table cnl (n-l-1) (k-1)
  return $ flip mod modN $ modN + x - y