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 Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.PrintWriter;
import java.util.*;

public class Main {
	static final long MOD = 1000000007;
	static long[] rev2;
	
	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		PrintWriter pw = new PrintWriter(System.out);
		int B = sc.nextInt();
		int W = sc.nextInt();
		
		rev2 = new long[B+W+1];
		rev2[0] = 1;
		rev2[1] = rev(2);
		for(int i=2; i<=B+W; i++)
			rev2[i] = (rev2[i-1]*rev2[1])%MOD;
		
		long[] combB = createComb(B+W, B-1);
		long[] combW= createComb(B+W, W-1);
		
		long[] pB = new long[B+W+1];
		long[] pW = new long[B+W+1];
		
		for(int i=0; i<B+W; i++) {
			long ans = ((1-pB[i]-pW[i]+MOD+MOD)*rev2[1] + pW[i])%MOD;
			pw.println(ans);
			pB[i+1] = (pB[i] + combB[i]*rev2[i+1])%MOD;
			pW[i+1] = (pW[i] + combW[i]*rev2[i+1])%MOD;
		}
		
		sc.close();
		pw.close();
	}
	
	static long[] createComb(int N, int v) {
		long[] ans = new long[N+1];
		ans[v] = 1;
		for(int i=v; i<N; i++)
			ans[i+1] = (((ans[i]*(i+1))%MOD)*rev(i+1-v))%MOD;
		
		return ans;
	}
	
	static long rev(long a) {
		return pow(a, MOD-2);
	}
	
	static long pow(long a, long b) {
		long ans = 1;
		while(b>0) {
			if((b&1)!=0)
				ans = (ans * a)%MOD;
			b>>=1;
			a = (a*a)%MOD;
		}
		return ans;
	}

}