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: Accepted)

using System;
using System.Collections.Generic;
class Program
{
	static void Main(string[] args)
	{
		string[] str = Console.ReadLine().Split();
		long B = long.Parse(str[0]);
		long W = long.Parse(str[1]);
		long mod = 1000000007;
		long[]C = new long[200001];
		long[]NC = new long[200001];
        C[0] = 1;
		C[1] = 1;
		NC[0] = 1;
		NC[1] = 1;
		for(var i=2;i<=200000;i++){
			C[i] = C[i-1] * i % mod;
			NC[i] = NC[i-1] * Mpow(i,mod-2,mod) % mod;
		}
		long[]A1 = new long[B+W];
		long[]A2 = new long[B+W];
		long c1 = 1;
		long p1 = 0;
		long q1 = 0;
		for(var i=0;i<B+W;i++){
			A1[i] = c1;
			A2[i] = c1*2%mod;
			if(i>=W){
				q1 = (q1 * 2 + C[i-1] * NC[i-W] % mod * NC[W-1] % mod) % mod;
				A1[i] = (c1 + q1) % mod;
			}
			if(i>=B){
				p1 = (p1 * 2 + C[i-1] * NC[i-B] % mod * NC[B-1] % mod) % mod;
				A1[i] = (A1[i] + mod - p1) % mod;
			}
			c1 = c1 * 2 % mod;
		}
		for(var i=0;i<B+W;i++){
			Console.WriteLine(A1[i] * Mpow(A2[i],mod-2,mod) % mod);
		}
	}
	public static long Mpow(long x,long n,long m)
	{
		long ans = 1;
		while(n > 0){
			if(n % 2 == 1){
				ans = ans * x % m;
				n -= 1;
			}
			x = x * x % m;
			n /= 2;
		}
		return ans;
	}
	public static long Ncr(long n,long r,long m)
	{
		long ans = 1;
		for(var i=n-r+1;i<=n;i++){
			ans = ans * i % m;
		}
		for(var i=2;i<=r;i++){
			ans = ans * Mpow(i,m-2,m) % m;
		}
		return ans;
	}
}