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)

use self::mod_int::ModInt;
use std::cmp;
use std::collections::{BTreeMap, BTreeSet, VecDeque};

const MOD: usize = 1e9 as usize + 7;
const INF: i64 = 1 << 60;

fn main() {
    let (r, w) = (std::io::stdin(), std::io::stdout());
    let mut sc = IO::new(r.lock(), w.lock());

    let b: usize = sc.read();
    let w: usize = sc.read();
    let n = b + w;
    let c = Combination::new(n + 1, MOD);
    let mut pow2 = vec![ModInt(1); n + 1];
    pow2[1] = ModInt(1) / 2;
    for i in 2..(n + 1) {
        pow2[i] = pow2[i - 1] * pow2[1];
    }

    let mut w_zero = vec![ModInt(0); n];
    for i in w..n {
        let b = i - w;
        w_zero[i] = c.get(i - 1, b) * pow2[i];
    }
    let mut b_zero = vec![ModInt(0); n];
    for i in b..n {
        let w = i - b;
        b_zero[i] = c.get(i - 1, w) * pow2[i];
    }
    let mut b_zero_sum = vec![ModInt(0); n + 1];
    for i in 0..n {
        b_zero_sum[i + 1] = b_zero_sum[i] + b_zero[i];
    }

    let mut w_zero_sum = vec![ModInt(0); n + 1];
    for i in 0..n {
        w_zero_sum[i + 1] = w_zero_sum[i] + w_zero[i];
    }

    for i in 0..n {
        let prev_b_zero = b_zero_sum[i + 1];
        let prev_w_zero = w_zero_sum[i + 1];
        let ans = (ModInt(1) - prev_b_zero - prev_w_zero) / 2 + prev_w_zero;
        println!("{}", ans.0);
    }
}

pub struct Combination {
    fact: Vec<usize>,
    inv_fact: Vec<usize>,
    modulo: usize,
}

impl Combination {
    pub fn new(max: usize, modulo: usize) -> Combination {
        let mut inv = vec![0; max + 1];
        let mut fact = vec![0; max + 1];
        let mut inv_fact = vec![0; max + 1];
        inv[1] = 1;
        for i in 2..(max + 1) {
            inv[i] = inv[modulo % i] * (modulo - modulo / i) % modulo;
        }
        fact[0] = 1;
        inv_fact[0] = 1;
        for i in 0..max {
            fact[i + 1] = fact[i] * (i + 1) % modulo;
        }
        for i in 0..max {
            inv_fact[i + 1] = inv_fact[i] * inv[i + 1] % modulo;
        }
        Combination {
            fact: fact,
            inv_fact: inv_fact,
            modulo: modulo,
        }
    }

    pub fn get(&self, x: usize, y: usize) -> ModInt<usize> {
        assert!(x >= y);
        ModInt(self.fact[x] * self.inv_fact[y] % self.modulo * self.inv_fact[x - y] % self.modulo)
    }
}

pub mod mod_int {
    use super::MOD;
    use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Sub, SubAssign};

    type Num = usize;

    #[derive(Clone, Copy)]
    pub struct ModInt<T: Copy + Clone>(pub T);

    impl Add<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self + rhs.0
        }
    }

    impl Add<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            let mut t = rhs + self.0;
            if t >= MOD {
                t = t - MOD;
            }
            ModInt(t)
        }
    }

    impl Sub<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: Num) -> ModInt<Num> {
            let rhs = if rhs >= MOD { rhs % MOD } else { rhs };
            let value = if self.0 < rhs { self.0 + MOD } else { self.0 };
            ModInt(value - rhs)
        }
    }

    impl Sub<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self - rhs.0
        }
    }

    impl AddAssign<Num> for ModInt<Num> {
        fn add_assign(&mut self, other: Num) {
            *self = *self + other;
        }
    }
    impl AddAssign<ModInt<Num>> for ModInt<Num> {
        fn add_assign(&mut self, other: ModInt<Num>) {
            *self = *self + other;
        }
    }

    impl SubAssign<Num> for ModInt<Num> {
        fn sub_assign(&mut self, other: Num) {
            *self = *self - other;
        }
    }

    impl SubAssign<ModInt<Num>> for ModInt<Num> {
        fn sub_assign(&mut self, other: ModInt<Num>) {
            *self = *self - other;
        }
    }

    impl Div<Num> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            self * ModInt(rhs).pow(MOD - 2)
        }
    }

    impl Div<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self / rhs.0
        }
    }

    impl DivAssign<Num> for ModInt<Num> {
        fn div_assign(&mut self, rhs: Num) {
            *self = *self / rhs
        }
    }
    impl DivAssign<ModInt<Num>> for ModInt<Num> {
        fn div_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self / rhs
        }
    }

    impl Mul<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, rhs: ModInt<Num>) -> ModInt<Num> {
            self * rhs.0
        }
    }
    impl Mul<Num> for ModInt<Num> {
        type Output = ModInt<Num>;

        fn mul(self, mut rhs: Num) -> ModInt<Num> {
            if rhs >= MOD {
                rhs %= MOD;
            }
            let t = (self.0 * rhs) % MOD;
            ModInt(t)
        }
    }

    impl MulAssign<Num> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: Num) {
            *self = *self * rhs;
        }
    }

    impl MulAssign<ModInt<Num>> for ModInt<Num> {
        fn mul_assign(&mut self, rhs: ModInt<Num>) {
            *self = *self * rhs;
        }
    }

    impl ModInt<Num> {
        pub fn pow(self, e: Num) -> ModInt<Num> {
            let mut result = ModInt(1);
            let mut cur = self;
            let mut e = e;
            while e > 0 {
                if e & 1 == 1 {
                    result *= cur;
                }
                e >>= 1;
                cur *= cur;
            }
            result
        }
    }
}

pub struct IO<R, W: std::io::Write>(R, std::io::BufWriter<W>);

impl<R: std::io::Read, W: std::io::Write> IO<R, W> {
    pub fn new(r: R, w: W) -> Self {
        IO(r, std::io::BufWriter::new(w))
    }
    pub fn write<S: ToString>(&mut self, s: S) {
        use std::io::Write;
        self.1.write(s.to_string().as_bytes()).unwrap();
    }
    pub fn read<T: std::str::FromStr>(&mut self) -> T {
        use std::io::Read;
        let buf = self
            .0
            .by_ref()
            .bytes()
            .map(|b| b.unwrap())
            .skip_while(|&b| b == b' ' || b == b'\n' || b == b'\r' || b == b'\t')
            .take_while(|&b| b != b' ' && b != b'\n' && b != b'\r' && b != b'\t')
            .collect::<Vec<_>>();
        unsafe { std::str::from_utf8_unchecked(&buf) }
            .parse()
            .ok()
            .expect("Parse error.")
    }
    pub fn vec<T: std::str::FromStr>(&mut self, n: usize) -> Vec<T> {
        (0..n).map(|_| self.read()).collect()
    }
    pub fn chars(&mut self) -> Vec<char> {
        self.read::<String>().chars().collect()
    }
}