Problem:
Score : 800 points

Problem Statement

There are N squares arranged in a row, numbered 1 to N from left to right. Takahashi will stack building blocks on these squares, on which there are no blocks yet.

He wants to stack blocks on the squares evenly, so he will repeat the following operation until there are H blocks on every square:

Let M and m be the maximum and minimum numbers of blocks currently stacked on a square, respectively. Choose a square on which m blocks are stacked (if there are multiple such squares, choose any one of them), and add a positive number of blocks on that square so that there will be at least M and at most M + D blocks on that square.

Tell him how many ways there are to have H blocks on every square by repeating this operation. Since there can be extremely many ways, print the number modulo 10^9+7.

Constraints

2 \leq N \leq 10^6

1 \leq D \leq H \leq 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N H D

Output

Print the number of ways to have H blocks on every square, modulo 10^9+7.

Sample Input 1

2 2 1

Sample Output 1

6

The possible transitions of (the number of blocks on Square 1, the number of blocks on Square 2) are as follows:

(0, 0) -> (0, 1) -> (1, 1) -> (1, 2) -> (2, 2)

(0, 0) -> (0, 1) -> (1, 1) -> (2, 1) -> (2, 2)

(0, 0) -> (0, 1) -> (2, 1) -> (2, 2)

(0, 0) -> (1, 0) -> (1, 1) -> (1, 2) -> (2, 2)

(0, 0) -> (1, 0) -> (1, 1) -> (2, 1) -> (2, 2)

(0, 0) -> (1, 0) -> (1, 2) -> (2, 2)

Thus, there are six ways to have two blocks on every square.

Sample Input 2

2 30 15

Sample Output 2

94182806

Sample Input 3

31415 9265 3589

Sample Output 3

312069529

Be sure to print the number modulo 10^9+7.

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

#[allow(unused_imports)]
use std::cmp::*;
#[allow(unused_imports)]
use std::collections::*;
use std::io::{Write, BufWriter};
// https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8
macro_rules! input {
    ($($r:tt)*) => {
        let stdin = std::io::stdin();
        let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock()));
        let mut next = move || -> String{
            bytes
                .by_ref()
                .map(|r|r.unwrap() as char)
                .skip_while(|c|c.is_whitespace())
                .take_while(|c|!c.is_whitespace())
                .collect()
        };
        input_inner!{next, $($r)*}
    };
}

macro_rules! input_inner {
    ($next:expr) => {};
    ($next:expr, ) => {};

    ($next:expr, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($next, $t);
        input_inner!{$next $($r)*}
    };
}

macro_rules! read_value {
    ($next:expr, ( $($t:tt),* )) => {
        ( $(read_value!($next, $t)),* )
    };

    ($next:expr, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    };

    ($next:expr, chars) => {
        read_value!($next, String).chars().collect::<Vec<char>>()
    };

    ($next:expr, usize1) => {
        read_value!($next, usize) - 1
    };

    ($next:expr, [ $t:tt ]) => {{
        let len = read_value!($next, usize);
        (0..len).map(|_| read_value!($next, $t)).collect::<Vec<_>>()
    }};

    ($next:expr, $t:ty) => {
        $next().parse::<$t>().expect("Parse error")
    };
}

/// Verified by https://atcoder.jp/contests/arc093/submissions/3968098
mod mod_int {
    use std::ops::*;
    pub trait Mod: Copy { fn m() -> i64; }
    #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)]
    pub struct ModInt<M> { pub x: i64, phantom: ::std::marker::PhantomData<M> }
    impl<M: Mod> ModInt<M> {
        // x >= 0
        pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) }
        fn new_internal(x: i64) -> Self {
            ModInt { x: x, phantom: ::std::marker::PhantomData }
        }
        pub fn pow(self, mut e: i64) -> Self {
            debug_assert!(e >= 0);
            let mut sum = ModInt::new_internal(1);
            let mut cur = self;
            while e > 0 {
                if e % 2 != 0 { sum *= cur; }
                cur *= cur;
                e /= 2;
            }
            sum
        }
        #[allow(dead_code)]
        pub fn inv(self) -> Self { self.pow(M::m() - 2) }
    }
    impl<M: Mod, T: Into<ModInt<M>>> Add<T> for ModInt<M> {
        type Output = Self;
        fn add(self, other: T) -> Self {
            let other = other.into();
            let mut sum = self.x + other.x;
            if sum >= M::m() { sum -= M::m(); }
            ModInt::new_internal(sum)
        }
    }
    impl<M: Mod, T: Into<ModInt<M>>> Sub<T> for ModInt<M> {
        type Output = Self;
        fn sub(self, other: T) -> Self {
            let other = other.into();
            let mut sum = self.x - other.x;
            if sum < 0 { sum += M::m(); }
            ModInt::new_internal(sum)
        }
    }
    impl<M: Mod, T: Into<ModInt<M>>> Mul<T> for ModInt<M> {
        type Output = Self;
        fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) }
    }
    impl<M: Mod, T: Into<ModInt<M>>> AddAssign<T> for ModInt<M> {
        fn add_assign(&mut self, other: T) { *self = *self + other; }
    }
    impl<M: Mod, T: Into<ModInt<M>>> SubAssign<T> for ModInt<M> {
        fn sub_assign(&mut self, other: T) { *self = *self - other; }
    }
    impl<M: Mod, T: Into<ModInt<M>>> MulAssign<T> for ModInt<M> {
        fn mul_assign(&mut self, other: T) { *self = *self * other; }
    }
    impl<M: Mod> Neg for ModInt<M> {
        type Output = Self;
        fn neg(self) -> Self { ModInt::new(0) - self }
    }
    impl<M> ::std::fmt::Display for ModInt<M> {
        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
            self.x.fmt(f)
        }
    }
    impl<M: Mod> ::std::fmt::Debug for ModInt<M> {
        fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result {
            let (mut a, mut b, _) = red(self.x, M::m());
            if b < 0 {
                a = -a;
                b = -b;
            }
            write!(f, "{}/{}", a, b)
        }
    }
    impl<M: Mod> From<i64> for ModInt<M> {
        fn from(x: i64) -> Self { Self::new(x) }
    }
    // Finds the simplest fraction x/y congruent to r mod p.
    // The return value (x, y, z) satisfies x = y * r + z * p.
    fn red(r: i64, p: i64) -> (i64, i64, i64) {
        if r.abs() <= 10000 {
            return (r, 1, 0);
        }
        let mut nxt_r = p % r;
        let mut q = p / r;
        if 2 * nxt_r >= r {
            nxt_r -= r;
            q += 1;
        }
        if 2 * nxt_r <= -r {
            nxt_r += r;
            q -= 1;
        }
        let (x, z, y) = red(nxt_r, r);
        (x, y - q * z, z)
    }
} // mod mod_int

macro_rules! define_mod {
    ($struct_name: ident, $modulo: expr) => {
        #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)]
        struct $struct_name {}
        impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } }
    }
}
const MOD: i64 = 1_000_000_007;
define_mod!(P, MOD);
type ModInt = mod_int::ModInt<P>;

// Depends on ModInt.rs
fn fact_init(w: usize) -> (Vec<ModInt>, Vec<ModInt>) {
    let mut fac = vec![ModInt::new(1); w];
    let mut invfac = vec![0.into(); w];
    for i in 1 .. w {
        fac[i] = fac[i - 1] * i as i64;
    }
    invfac[w - 1] = fac[w - 1].inv();
    for i in (0 .. w - 1).rev() {
        invfac[i] = invfac[i + 1] * (i as i64 + 1);
    }
    (fac, invfac)
}

fn solve() {
    let out = std::io::stdout();
    let mut out = BufWriter::new(out.lock());
    macro_rules! puts {
        ($($format:tt)*) => (write!(out,$($format)*).unwrap());
    }
    input!(n: usize, h: usize, d: usize);
    const W: usize = 3_000_000;
    let (fac, invfac) = fact_init(W);
    let mut dp = vec![ModInt::new(0); h + 1];
    dp[0] += fac[n];
    let mut acc = vec![ModInt::new(0); h + 2];
    acc[0] += fac[n];
    let mut facsum = ModInt::new(0);
    for i in 1..n + 1 {
        facsum += fac[i];
    }
    for i in 1..h + 1 {
        if i <= d {
            dp[i] += fac[n];
        }
        let lo = max(d + 1, i) - d;
        let hi = i;
        dp[i] += (acc[hi] - acc[lo]) * facsum;
        acc[i + 1] = acc[i] + dp[i];
    }
    
    puts!("{}\n", dp[h]);
}

fn main() {
    // In order to avoid potential stack overflow, spawn a new thread.
    let stack_size = 104_857_600; // 100 MB
    let thd = std::thread::Builder::new().stack_size(stack_size);
    thd.spawn(|| solve()).unwrap().join().unwrap();
}