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 Python (3.4.3) (judge verdict: Time Limit Exceeded)

B, W = map(int, input().split(" "))
MOD = 10 ** 9 + 7

# x * z - MOD * n == y
def calcz(x, y):
    a, b = x, MOD
    syou = []
    while a > 1:
        syou.append(b // a)
        a, b = b % a, a
    syou.reverse()
    dp = [0, 1]
    for i in range(len(syou)):
        dp.append(dp[-1] * (-syou[i]) + dp[-2])
    return (dp[-1] * y) % MOD

p_bl = [0 for i in range(B+W)]
p_wh = [0 for i in range(B+W)]
bunbo = [1]

btmp = 1
wtmp = 1
print(500000004)
for i in range(1, B + W):
    bunbo.append(bunbo[-1] * 2 % MOD)
    if i >= B:
        p_bl[i] = (p_bl[i-1] * 2 + btmp) % MOD
        btmp = btmp * i // (i - B + 1)
        
    if i >= W:
        p_wh[i] = (p_wh[i-1] * 2 + wtmp) % MOD
        wtmp = wtmp * i // (i - W + 1)
        
    if i < min(B, W):
        print(500000004)
    else:
        bunshi = p_wh[i] + (bunbo[i] - p_bl[i])
        print(calcz(bunbo[i] * 2, bunshi))