Problem:
Score : 700 points

Problem Statement

Today, Snuke will eat B pieces of black chocolate and W pieces of white chocolate for an afternoon snack.

He will repeat the following procedure until there is no piece left:

Choose black or white with equal probability, and eat a piece of that color if it exists.

For each integer i from 1 to B+W (inclusive), find the probability that the color of the i-th piece to be eaten is black.
It can be shown that these probabilities are rational, and we ask you to print them modulo 10^9 + 7, as described in Notes.

Notes

When you print a rational number, first write it as a fraction \frac{y}{x}, where x, y are integers and x is not divisible by 10^9 + 7
(under the constraints of the problem, such representation is always possible).
Then, you need to print the only integer z between 0 and 10^9 + 6, inclusive, that satisfies xz \equiv y \pmod{10^9 + 7}.

Constraints

All values in input are integers.

1 \leq B,W \leq 10^{5}

Input

Input is given from Standard Input in the following format:

B W

Output

Print the answers in B+W lines. In the i-th line, print the probability that the color of the i-th piece to be eaten is black, modulo 10^{9}+7.

Sample Input 1

2 1

Sample Output 1

500000004
750000006
750000006

There are three possible orders in which Snuke eats the pieces:

white, black, black

black, white, black

black, black, white

with probabilities \frac{1}{2}, \frac{1}{4}, \frac{1}{4}, respectively. Thus, the probabilities of eating a black piece first, second and third are \frac{1}{2},\frac{3}{4} and \frac{3}{4}, respectively.

Sample Input 2

3 2

Sample Output 2

500000004
500000004
625000005
187500002
187500002

They are \frac{1}{2},\frac{1}{2},\frac{5}{8},\frac{11}{16} and \frac{11}{16}, respectively.

Sample Input 3

6 9

Sample Output 3

500000004
500000004
500000004
500000004
500000004
500000004
929687507
218750002
224609377
303710940
633300786
694091802
172485353
411682132
411682132

Solution in Haskell (GHC 7.10.3) (judge verdict: Runtime Error)

{-# LANGUAGE BangPatterns #-}
import Control.Monad
import Data.Int
import qualified Data.Vector.Unboxed as U
import qualified Data.Vector.Unboxed.Mutable as UM
import qualified Data.ByteString.Char8 as BS
import Data.Array.IO

exEuclid :: (Eq a, Integral a) => a -> a -> (a, a, a)
exEuclid f g = loop 1 0 0 1 f g
  where loop u0 u1 v0 v1 f 0 = (f, u0, v0)
        loop u0 u1 v0 v1 f g =
          case divMod f g of
            (q,r) -> loop u1 (u0 - q * u1) v1 (v0 - q * v1) g r

modulo = 10^9+7 :: Int64
addMod, subMod, mulMod, divM :: Int64 -> Int64 -> Int64
addMod x y = (x + y) `rem` modulo
subMod x y = (x - y) `mod` modulo
mulMod x y = (x * y) `rem` modulo
recipM :: Int64 -> Int64
recipM x = case exEuclid x modulo of
             (1,a,_) -> a `mod` modulo
             (-1,a,_) -> (-a) `mod` modulo
divM x y = x `mulMod` recipM y

newtype N = N Int64 deriving (Eq, Show)
instance Num N where
  N x + N y = N ((x + y) `rem` modulo)
  N x - N y = N ((x - y) `mod` modulo)
  N x * N y = N ((x * y) `rem` modulo)
  fromInteger n = N (fromInteger (n `mod` fromIntegral modulo))
  abs = undefined; signum = undefined

instance Fractional N where
  N x / N y = N (divM x y)
  recip (N x) = N (recipM x)
  fromRational = undefined

main = do
  [b,w] <- map (read . BS.unpack) . BS.words <$> BS.getLine
  let half = 1 / 2 :: N
  arr <- newArray ((0,0),(b+1,w+1)) 0 :: IO (IOUArray (Int,Int) Int64)
  writeArray arr (b,w) 1
  vec <- UM.new (b+w) :: IO (UM.IOVector Int64)
  forM_ [b+w-1,b+w-2..0] $ \i -> do
    forM_ [max 0 (i - w)..min b i] $ \j -> do
      let k = i - j
      -- j + k == i
      -- 0 <= j <= b, 0 <= k <= w
      u <- readArray arr (j+1,k)
      v <- readArray arr (j,k+1)
      let N uu | k == 0 = N u
               | otherwise = half * N u
      let N vv | j == 0 = N v
               | otherwise = half * N v
      UM.modify vec (`addMod` uu) i
      writeArray arr (j,k) (uu `addMod` vv)
  {-
  forM_ [0..b] $ \j -> do
    r <- forM [0..w] $ \k -> readArray arr (j,k)
    print r
  -}
  forM_ [b+w-1,b+w-2..0] $ \i -> do
    print =<< UM.read vec i