Problem:
Score: 1000 points

Problem Statement

Find the number of sequences of N non-negative integers A_1, A_2, ..., A_N that satisfy the following conditions:

L \leq A_1 + A_2 + ... + A_N \leq R

When the N elements are sorted in non-increasing order, the M-th and (M+1)-th elements are equal.

Since the answer can be enormous, print it modulo 10^9+7.

Constraints

All values in input are integers.

1 \leq M < N \leq 3 \times 10^5

1 \leq L \leq R \leq 3 \times 10^5

Input

Input is given from Standard Input in the following format:

N M L R

Output

Print the number of sequences of N non-negative integers, modulo 10^9+7.

Sample Input 1

4 2 3 7

Sample Output 1

105

The sequences of non-negative integers that satisfy the conditions are:

\begin{eqnarray} &(1, 1, 1, 0), (1, 1, 1, 1), (2, 1, 1, 0), (2, 1, 1, 1), (2, 2, 2, 0), (2, 2, 2, 1), \\ &(3, 0, 0, 0), (3, 1, 1, 0), (3, 1, 1, 1), (3, 2, 2, 0), (4, 0, 0, 0), (4, 1, 1, 0), \\ &(4, 1, 1, 1), (5, 0, 0, 0), (5, 1, 1, 0), (6, 0, 0, 0), (7, 0, 0, 0)\end{eqnarray}

and their permutations, for a total of 105 sequences.

Sample Input 2

2 1 4 8

Sample Output 2

3

The three sequences that satisfy the conditions are (2, 2), (3, 3), and (4, 4).

Sample Input 3

141592 6535 89793 238462

Sample Output 3

933832916

Solution in Rust (1.15.1) (judge verdict: Accepted)

// ---------- begin ModInt ----------
const MOD: u32 = 1_000_000_007;

#[derive(Clone, Copy)]
struct ModInt(u32);

impl std::ops::Add for ModInt {
    type Output = ModInt;
    fn add(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + rhs.0;
        if d >= MOD {
            d -= MOD;
        }
        ModInt(d)
    }
}

impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: ModInt) {
        *self = *self + rhs;
    }
}

impl std::ops::Sub for ModInt {
    type Output = ModInt;
    fn sub(self, rhs: ModInt) -> Self::Output {
        let mut d = self.0 + MOD - rhs.0;
        if d >= MOD {
            d -= MOD;
        }
        ModInt(d)
    }
}

impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: ModInt) {
        *self = *self - rhs;
    }
}

impl std::ops::Mul for ModInt {
    type Output = ModInt;
    fn mul(self, rhs: ModInt) -> Self::Output {
        ModInt((self.0 as u64 * rhs.0 as u64 % MOD as u64) as u32)
    }
}

impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: ModInt) {
        *self = *self * rhs;
    }
}

impl std::ops::Neg for ModInt {
    type Output = ModInt;
    fn neg(self) -> Self::Output {
        ModInt(if self.0 == 0 {0} else {MOD - self.0})
    }
}

impl std::fmt::Display for ModInt {
    fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl std::str::FromStr for ModInt {
    type Err = std::num::ParseIntError;
    fn from_str(s: &str) -> Result<Self, Self::Err> {
        let val = s.parse::<u32>()?;
        Ok(ModInt::new(val))
    }
}

#[allow(dead_code)]
impl ModInt {
    pub fn new(n: u32) -> ModInt {
        ModInt(n % MOD)
    }
    pub fn zero() -> ModInt {
        ModInt(0)
    }
    pub fn one() -> ModInt {
        ModInt(1)
    }
    pub fn pow(self, mut n: u32) -> ModInt {
        let mut t = ModInt::one();
        let mut s = self;
        while n > 0 {
            if n & 1 == 1 {
                t *= s;
            }
            s *= s;
            n >>= 1;
        }
        t
    }
    pub fn inv(self) -> ModInt {
        self.pow(MOD - 2)
    }
}
// ---------- end ModInt ----------
// ---------- begin Precalc ----------
#[allow(dead_code)]
struct Precalc {
    inv: Vec<ModInt>,
    fact: Vec<ModInt>,
    ifact: Vec<ModInt>,
}

#[allow(dead_code)]
impl Precalc {
    pub fn new(n: usize) -> Precalc {
        let mut inv = vec![ModInt::one(); n + 1];
        let mut fact = vec![ModInt::one(); n + 1];
        let mut ifact = vec![ModInt::one(); n + 1];
        for i in 2..(n + 1) {
            inv[i] = -inv[MOD as usize % i] * ModInt(MOD / i as u32);
            fact[i] = fact[i - 1] * ModInt(i as u32);
            ifact[i] = ifact[i - 1] * inv[i];
        }
        Precalc {
            inv: inv,
            fact: fact,
            ifact: ifact,
        }
    }
    pub fn inv(&self, n: usize) -> ModInt {
        self.inv[n]
    }
    pub fn fact(&self, n: usize) -> ModInt {
        self.fact[n]
    }
    pub fn ifact(&self, n: usize) -> ModInt {
        self.ifact[n]
    }
    pub fn comb(&self, n: usize, k: usize) -> ModInt {
        if k > n {
            return ModInt::zero();
        }
        self.fact[n] * self.ifact[k] * self.ifact[n - k]
    }
}
// ---------- end Precalc ----------

use std::cmp::*;

fn run() {
    let mut s = String::new();
    std::io::stdin().read_line(&mut s).unwrap();
    let it = s.trim().split_whitespace();
    let a: Vec<usize> = it.map(|s| s.parse().unwrap()).collect();
    let (n, m, l, r) = (a[0], a[1], a[2], a[3]);
    let pc = Precalc::new(r + n);
    let calc = |x: usize| -> ModInt {
        let mut all = pc.comb(x + n, n);
        for k in 1..(x / m + 1) {
            let rem = x - k * m;
            let mut sign = ModInt::one();
            let mut way = ModInt::zero();
            for i in 0..(min(n - m, rem / k) + 1) {
                let mut v = pc.comb(rem - k * i + n, n);
                if rem >= k * i + m {
                    v -= pc.comb(rem - k * i - m + n, n);
                }
                way += sign * v * pc.comb(n - m, i);
                sign = -sign;
            }
            all -= way * pc.comb(n, m);
        }
        all
    };
    let ans = calc(r) - calc(l - 1);
    println!("{}", ans);
}

fn main() {
    run();
}