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 PyPy3 (2.4.0) (judge verdict: Time Limit Exceeded)

B, W = map(int,input().split())
Mod = 10**9+7
magick = 500000004

DP_1 = [[0 for j in range(B+1)] for i in range(B+W+1)]
DP_1[0][B] = 1
for i in range(B+W):
  for j in range(B+1):
    if j == 0:
      DP_1[i+1][j] = 2*DP_1[i][j]+DP_1[i][j+1]
    elif j == B:
      DP_1[i+1][j] = DP_1[i][j]
    else:
      DP_1[i+1][j] = DP_1[i][j] +DP_1[i][j+1]
      
DP_2 = [[0 for j in range(W+1)] for i in range(B+W+1)]
DP_2[0][W] = 1
for i in range(B+W):
  for j in range(W+1):
    if j == 0:
      DP_2[i+1][j] = 2*DP_2[i][j]+DP_2[i][j+1]
    elif j == W:
      DP_2[i+1][j] = DP_2[i][j]
    else:
      DP_2[i+1][j] = DP_2[i][j] +DP_2[i][j+1]
      
inv = 1
for i in range(B+W):
  print(((-DP_1[i][0]+DP_2[i][0])*inv*magick+ magick)%Mod)
  inv = inv * magick%Mod