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 C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
class Program
    {
        static long mod = 1000000007;
        static long sq(long i){
            return times(i, i) % mod;
        }

        static long times(long a, long b)
        {
            return a * b % mod;
        }

        static long pow(long a, long n)
        {
            int p = 0;
            long ret = 1;
            long buf = a;
            while ((1 << p) <= n)
            {
                if(((n >> p) & 1) > 0)
                {
                    ret = times(ret, buf);
                }
                buf = sq(buf);
                p++;
            }
            return ret;
        }

        static long inv (long a)
        {
            return pow(a, mod - 2);
        }

    	static void Main(string[] args)
    	{
            string[] s = Console.ReadLine().Split(' ');
            long b = long.Parse(s[0]), w = long.Parse(s[1]);

            long black = 0, white = 0;
            long nCb = 1, nCw = 1;

            for (int i=0; i<b+w; i++)
            {

                if (i < b) ;
                else if (i == b) black = inv(pow(2, b));
                else
                {
                    nCb = nCb * (i - 1) % mod * inv(i - b) % mod;
                    black = (black + (nCb % mod * inv(pow(2, i))) % mod) % mod;
                }
                if (i < w) ;
                else if (i == w) white = inv(pow(2, w));
                else
                {
                    nCw = nCw * (i - 1) % mod * inv(i - w) % mod;
                    white = (white + (nCw % mod * inv(pow(2, i))) % mod) % mod;
                }

                long ans = 1 * inv(2) % mod;
                ans += white * inv(2) % mod;
                ans -= black * inv(2) % mod;
                ans = ans % mod;
                if (ans < 0) ans += mod;
                //Console.WriteLine("white: " + white);
                //Console.WriteLine("black: " + black);
                Console.WriteLine(ans);
            }
                        
        }
    }