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 Rust (1.15.1) (judge verdict: Accepted)

// ---------- begin ModInt ----------

//https://github.com/kenkoooo/competitive-programming-rs/blob/master/src/math/mod_int.rs
//を参考にしています
#[allow(dead_code)]
mod modint {
    pub const MOD: u32 = 1000000000 + 7;
    use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};
    #[derive(Clone, Copy)]
    pub struct ModInt<T: Copy + Clone>(pub T);
    type Num = u32;
    impl Add<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, other: ModInt<Num>) -> ModInt<Num> {
            let mut d = self.0 + other.0;
            if d >= MOD {
                d -= MOD;
            }
            ModInt(d)
        }
    }
    impl AddAssign<ModInt<Num>> for ModInt<Num> {
        fn add_assign(&mut self, other: ModInt<Num>) {
            *self = *self + other;
        }
    }
    impl Sub<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, other: ModInt<Num>) -> ModInt<Num> {
            let mut d = self.0 + MOD - other.0;
            if d >= MOD {
                d -= MOD;
            }
            ModInt(d)
        }
    }
    impl SubAssign<ModInt<Num>> for ModInt<Num> {
        fn sub_assign(&mut self, other: ModInt<Num>) {
            *self = *self - other;
        }
    }
    impl Mul<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn mul(self, other: ModInt<Num>) -> ModInt<Num> {
            ModInt(((self.0 as u64) * (other.0 as u64) % (MOD as u64)) as u32)
        }
    }
    impl MulAssign<ModInt<Num>> for ModInt<Num> {
        fn mul_assign(&mut self, other: ModInt<Num>) {
            *self = *self * other;
        }
    }
    impl Div<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, other: Num) -> ModInt<Num> {
            self * ModInt(other).pow(MOD - 2)
        }
    }
    impl Div<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, other: ModInt<Num>) -> ModInt<Num> {
            self * other.pow(MOD - 2)
        }
    }
    impl DivAssign<ModInt<Num>> for ModInt<Num> {
        fn div_assign(&mut self, other: ModInt<Num>) {
            *self = *self / other;
        }
    }
    impl Neg for ModInt<Num> {
        type Output = ModInt<Num>;
        fn neg(self) -> ModInt<Num> {
            ModInt(if self.0 == 0 { 0 } else { MOD - self.0 })
        }
    }
    impl ModInt<Num> {
        pub fn new(v: u32) -> ModInt<Num> {
            ModInt(v % MOD)
        }
        pub fn pow(self, mut n: u32) -> ModInt<Num> {
            let mut t = ModInt::new(1);
            let mut s = self;
            while n > 0 {
                if (n & 1) == 1 {
                    t *= s;
                }
                s *= s;
                n >>= 1;
            }
            t
        }
    }
    pub struct Precalc {
        n: usize,
        inv: Vec<ModInt<Num>>,
        fact: Vec<ModInt<Num>>,
        ifact: Vec<ModInt<Num>>,
    }
    impl Precalc {
        pub fn new(n: usize) -> Precalc {
            let mut inv = vec![ModInt(1); n + 1];
            let mut fact = vec![ModInt(1); n + 1];
            let mut ifact = vec![ModInt(1); n + 1];
            for i in 1..(n + 1) {
                if i >= 2 {
                    inv[i] = -inv[(MOD as usize) % i] * ModInt(MOD / (i as u32));
                }
                fact[i] = ModInt(i as u32) * fact[i - 1];
                ifact[i] = inv[i] * ifact[i - 1];
            }
            Precalc {
                n: n,
                inv: inv,
                fact: fact,
                ifact: ifact,
            }
        }
        pub fn fact(&self, n: usize) -> ModInt<Num> {
            self.fact[n]
        }
        pub fn inv(&self, x: usize) -> ModInt<Num> {
            self.inv[x]
        }
        pub fn ifact(&self, x: usize) -> ModInt<Num> {
            self.ifact[x]
        }
        pub fn comb(&self, n: usize, k: usize) -> ModInt<Num> {
            if !(k <= n) {
                return ModInt(0);
            }
            self.fact[n] * self.ifact[k] * self.ifact[n - k]
        }
    }
    use std;
    impl std::fmt::Display for ModInt<Num> {
        fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
            write!(f, "{}", self.0)
        }
    }
}

// ---------- end ModInt ----------

use std::io::Write;
use modint::*;

fn run() {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).unwrap();
    let mut it = s.split_whitespace();
    let b: usize = it.next().unwrap().parse().unwrap();
    let w: usize = it.next().unwrap().parse().unwrap();
    let out = std::io::stdout();
    let mut out = std::io::BufWriter::new(out.lock());
    let pc = Precalc::new(b + w);
    let mut x = ModInt(0);
    let mut y = ModInt(0);
    for i in 1..(b + w + 1) {
        let ans = y + (ModInt(1) - x - y) * pc.inv(2);
        writeln!(out, "{}", ans).unwrap();
        if i >= b {
            x += pc.comb(i - 1, b - 1) * ModInt::pow(pc.inv(2), i as u32);
        }
        if i >= w {
            y += pc.comb(i - 1, w - 1) * ModInt::pow(pc.inv(2), i as u32);
        }
    }
}

fn main() {
    run();
}