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;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Text;
using System.Threading.Tasks;
using static System.Console;
using static System.Math;
class Z { static void Main() => new K(); }
class K
{
	int F => int.Parse(Str);
	long FL => long.Parse(Str);
	int[] G => Strs.Select(int.Parse).ToArray();
	uint[] GU => Strs.Select(uint.Parse).ToArray();
	decimal[] GD => Strs.Select(decimal.Parse).ToArray();
	long[] GL => Strs.Select(long.Parse).ToArray();
	string Str => ReadLine();
	string[] Strs => Str.Split(new char[] { ' ', '\t' }, StringSplitOptions.RemoveEmptyEntries);
	static T[] ConstantArray<T>(int n, T val) { var a = new T[n]; for (var i = 0; i < n; i++) a[i] = val; return a; }
	static T[] MakeArray<T>(int n, Func<int, T> f) { var a = new T[n]; for (var i = 0; i < n; i++) a[i] = f(i); return a; }
	static T[] Recursion<T>(int n, T v, Func<int, T, T> f) { var a = new T[n]; a[0] = v; for (var i = 1; i < n; i++) a[i] = f(i, a[i - 1]); return a; }
	static ulong One(int k) => 1ul << k;
	static ulong All(int k) => One(k) - 1;
	static bool Has(ulong s, int k) => (s & One(k)) != 0;
	const int MOD = 1000000007;
	public K()
	{
		SetOut(new StreamWriter(OpenStandardOutput()) { AutoFlush = false });
		Solve();
		Out.Flush();
	}
	static readonly ModInt Z = ModInt.Two.Inverse;
	ModInt[] f;
	void Solve()
	{
		var I = G;
		int B = I[0], W = I[1];
		f = Recursion(B + W + 1, ModInt.One, (i, x) => i * x);
		var p = Hoge(B, W);
		var q = Hoge(W, B);
		for (var i = 1; i <= B + W; i++)
		{
			var ans = ModInt.One;
			if (i >= W + 1) ans += p[i - W - 1];
			if (i >= B + 1) ans -= q[i - B - 1];
			WriteLine(ans / 2);
		}
	}
	public ModInt[] Hoge(int B, int W) => Recursion(B + 1, Z ^ W, (b, x) => (x + Binom(b + W - 1, b) * (Z ^ (b + W))));
	public ModInt Binom(int a, int b) => f[a] / f[b] / f[a - b];
	public static long Inverse(long a, long mod = MOD)
	{
		if (a < 0) { a %= mod; if (a < 0) a += mod; }
		var t = SolveLinear(a, mod);
		return t.Item1 > 0 ? t.Item1 : t.Item1 + mod;
	}
	public static Tuple<long, long> SolveLinear(long n, long m)
	{
		if (n < 0) { var p = SolveLinear(-n, m); return p == null ? p : new Tuple<long, long>(-p.Item1, p.Item2); }
		if (m < 0) { var p = SolveLinear(n, -m); return p == null ? p : new Tuple<long, long>(p.Item1, -p.Item2); }
		if (n < m) { var p = SolveLinear(m, n); return p == null ? p : new Tuple<long, long>(p.Item2, p.Item1); }
		long a = 1, b = 0, c = 0, d = 1;
		while (m > 0)
		{
			var r = n % m;
			var q = n / m;
			n = m;
			m = r;
			var tmp = a;
			a = -a * q + b;
			b = tmp;
			tmp = c;
			c = -c * q + d;
			d = tmp;
		}
		return n != 1 ? null : new Tuple<long, long>(d, b);
	}
}
struct ModInt : IEquatable<ModInt>
{
	public static readonly ModInt Zero = 0;
	public static readonly ModInt One = 1;
	public static readonly ModInt Two = 2;
	const int MOD = 1000000007;
	public readonly int Val;
	public ModInt(long x) { Val = (int)(x % MOD); }
	public ModInt Inverse => new ModInt(K.Inverse(Val));
	public static ModInt operator +(ModInt a) => new ModInt(a.Val);
	public static ModInt operator -(ModInt a) => new ModInt(MOD - a.Val);
	public static ModInt operator +(ModInt a, ModInt b) => new ModInt(a.Val + b.Val);
	public static ModInt operator -(ModInt a, ModInt b) => new ModInt(a.Val - b.Val + MOD);
	public static ModInt operator *(ModInt a, ModInt b) => new ModInt((long)a.Val * b.Val);
	public static ModInt operator /(ModInt a, ModInt b) => a * b.Inverse;
	public static ModInt operator ^(ModInt a, long b)
	{
		var p = ModInt.One;
		var x = a;
		while (b > 0)
		{
			if ((b & 1) == 1) p *= x;
			b >>= 1;
			x *= x;
		}
		return p;
	}
	public static implicit operator ModInt(long x) => new ModInt(x);
	public static explicit operator int(ModInt x) => x.Val;
	public override string ToString() => Val.ToString();
	public override int GetHashCode() => Val.GetHashCode();
	public override bool Equals(object obj) => Val == ((ModInt)obj).Val;
	public bool Equals(ModInt other) => Val == other.Val;
}