// ---------- 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(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result { write!(f, "{}", self.0) } } */ #[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) } pub fn comb(n: u32, k: u32) -> ModInt { if k > n { return ModInt::zero(); } let k = std::cmp::min(k, n - k); let mut nu = ModInt::one(); let mut de = ModInt::one(); for i in 0..k { nu *= ModInt(n - i); de *= ModInt(i + 1); } nu * de.inv() } } #[allow(dead_code)] struct Precalc { inv: Vec, fact: Vec, ifact: Vec, } #[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 ModInt ---------- use std::io::Read; fn karatsuba(c: &mut [ModInt], a: &[ModInt], b: &[ModInt]) { assert!(a.len() == b.len()); assert!(c.len() == 2 * a.len()); if a.len() < 64 { for (i, a) in a.iter().enumerate() { for (c, b) in c[i..].iter_mut().zip(b.iter()) { *c += *a * *b; } } return; } if a.len() & 1 == 1 { karatsuba(&mut c[2..], &a[1..], &b[1..]); let x = a[0]; let y = b[0]; c[0] += x * y; for (c, (a, b)) in c[1..].iter_mut().zip(a[1..].iter().zip(b[1..].iter())) { *c += x * *b + *a * y; } return; } let n = a.len(); let m = n / 2; karatsuba(&mut c[..n], &a[..m], &b[..m]); karatsuba(&mut c[n..], &a[m..], &b[m..]); let mut x = vec![ModInt::zero(); 2 * n]; let mut y = x.split_off(m); let mut z = y.split_off(m); for (x, (a, b)) in x.iter_mut().zip(a.iter().zip(a[m..].iter())) { *x = *a + *b; } for (y, (a, b)) in y.iter_mut().zip(b.iter().zip(b[m..].iter())) { *y = *a + *b; } karatsuba(&mut z, &x, &y); for (z, (a, b)) in z.iter_mut().zip(c.iter().zip(c[n..].iter())) { *z -= *a + *b; } for (c, z) in c[m..].iter_mut().zip(z.iter()) { *c += *z; } } fn inverse(a: &Vec, n: usize) -> Vec { assert!(a[0].0 > 0); let mut b = vec![a[0].inv(); 1]; let mut k = 1; while k < n { k *= 2; let mut c = vec![ModInt::zero(); k]; karatsuba(&mut c, &b, &b); let mut x = a.clone(); x.resize(k, ModInt::zero()); let mut y = vec![ModInt::zero(); 2 * k]; karatsuba(&mut y, &x, &c); b.resize(k, ModInt::zero()); for (b, y) in b.iter_mut().zip(y.iter()) { *b = *b + *b - *y; } } b } fn run() { let mut s = String::new(); std::io::stdin().read_to_string(&mut s).unwrap(); let mut it = s.trim().split_whitespace(); let k: usize = it.next().unwrap().parse().unwrap(); let n: usize = it.next().unwrap().parse().unwrap(); let mut a = vec![ModInt::zero(); 100_000 + 1]; a[0] = ModInt::one(); for _ in 0..n { let x: usize = it.next().unwrap().parse().unwrap(); a[x] = -ModInt::one(); } let inv = inverse(&a, k + 1); println!("{}", inv[k].0); } fn main() { run(); }