use std::io::Read; fn get_word() -> String { let stdin = std::io::stdin(); let mut stdin=stdin.lock(); let mut u8b: [u8; 1] = [0]; loop { let mut buf: Vec = Vec::with_capacity(16); loop { let res = stdin.read(&mut u8b); if res.unwrap_or(0) == 0 || u8b[0] <= b' ' { break; } else { buf.push(u8b[0]); } } if buf.len() >= 1 { let ret = String::from_utf8(buf).unwrap(); return ret; } } } fn get() -> T { get_word().parse().ok().unwrap() } /// Verified by https://atcoder.jp/contests/abc198/submissions/21774342 mod mod_int { use std::ops::*; pub trait Mod: Copy { fn m() -> i64; } #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)] pub struct ModInt { pub x: i64, phantom: ::std::marker::PhantomData } impl ModInt { // 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 Default for ModInt { fn default() -> Self { Self::new_internal(0) } } impl>> Add for ModInt { 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>> Sub for ModInt { 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>> Mul for ModInt { type Output = Self; fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) } } impl>> AddAssign for ModInt { fn add_assign(&mut self, other: T) { *self = *self + other; } } impl>> SubAssign for ModInt { fn sub_assign(&mut self, other: T) { *self = *self - other; } } impl>> MulAssign for ModInt { fn mul_assign(&mut self, other: T) { *self = *self * other; } } impl Neg for ModInt { type Output = Self; fn neg(self) -> Self { ModInt::new(0) - self } } impl ::std::fmt::Display for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { self.x.fmt(f) } } impl ::std::fmt::Debug for ModInt { 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 From for ModInt { 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 MInt = mod_int::ModInt

; // FFT (in-place, verified as NTT only) // R: Ring + Copy // Verified by: https://judge.yosupo.jp/submission/53831 // Adopts the technique used in https://judge.yosupo.jp/submission/3153. mod fft { use std::ops::*; // n should be a power of 2. zeta is a primitive n-th root of unity. // one is unity // Note that the result is bit-reversed. pub fn fft(f: &mut [R], zeta: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let mut m = n; let mut base = zeta; unsafe { while m > 2 { m >>= 1; let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m); *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = w * (u - d); w = w * base; } r += 2 * m; } base = base * base; } if m > 1 { // m = 1 let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } } } } pub fn inv_fft(f: &mut [R], zeta_inv: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let zeta = zeta_inv; // inverse FFT let mut zetapow = Vec::with_capacity(20); { let mut m = 1; let mut cur = zeta; while m < n { zetapow.push(cur); cur = cur * cur; m *= 2; } } let mut m = 1; unsafe { if m < n { zetapow.pop(); let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } m = 2; } while m < n { let base = zetapow.pop().unwrap(); let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m) * w; *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = u - d; w = w * base; } r += 2 * m; } m *= 2; } } } } mod arbitrary_mod { use crate::mod_int::{self, ModInt}; use crate::fft; const MOD1: i64 = 1012924417; const MOD2: i64 = 1224736769; const MOD3: i64 = 1007681537; const G1: i64 = 5; const G2: i64 = 3; const G3: i64 = 3; define_mod!(P1, MOD1); define_mod!(P2, MOD2); define_mod!(P3, MOD3); fn zmod(mut a: i64, b: i64) -> i64 { a %= b; if a < 0 { a += b; } a } fn ext_gcd(mut a: i64, mut b: i64) -> (i64, i64, i64) { let mut x = 0; let mut y = 1; let mut u = 1; let mut v = 0; while a != 0 { let q = b / a; x -= q * u; std::mem::swap(&mut x, &mut u); y -= q * v; std::mem::swap(&mut y, &mut v); b -= q * a; std::mem::swap(&mut b, &mut a); } (b, x, y) } fn invmod(a: i64, b: i64) -> i64 { let x = ext_gcd(a, b).1; zmod(x, b) } // This function is ported from http://math314.hateblo.jp/entry/2015/05/07/014908 fn garner(mut mr: Vec<(i64, i64)>, mo: i64) -> i64 { mr.push((mo, 0)); let mut coffs = vec![1; mr.len()]; let mut constants = vec![0; mr.len()]; for i in 0..mr.len() - 1 { let v = zmod(mr[i].1 - constants[i], mr[i].0) * invmod(coffs[i], mr[i].0) % mr[i].0; assert!(v >= 0); for j in i + 1..mr.len() { constants[j] += coffs[j] * v % mr[j].0; constants[j] %= mr[j].0; coffs[j] = coffs[j] * mr[i].0 % mr[j].0; } } constants[mr.len() - 1] } // f *= g, g is destroyed fn convolution_friendly(a: &[i64], b: &[i64], gen: i64) -> Vec { let d = a.len(); let mut f = vec![ModInt::

::new(0); d]; let mut g = vec![ModInt::

::new(0); d]; for i in 0..d { f[i] = a[i].into(); g[i] = b[i].into(); } let zeta = ModInt::new(gen).pow((P::m() - 1) / d as i64); fft::fft(&mut f, zeta, ModInt::new(1)); fft::fft(&mut g, zeta, ModInt::new(1)); for i in 0..d { f[i] *= g[i]; } fft::inv_fft(&mut f, zeta.inv(), ModInt::new(1)); let inv = ModInt::new(d as i64).inv(); let mut ans = vec![0; d]; for i in 0..d { ans[i] = (f[i] * inv).x; } ans } // Precondition: 0 <= a[i], b[i] < mo pub fn arbmod_convolution(a: &[i64], b: &[i64], mo: i64, ret: &mut [i64]) { use crate::mod_int::Mod; let d = a.len(); assert!(d.is_power_of_two()); assert_eq!(d, b.len()); let x = convolution_friendly::(&a, &b, G1); let y = convolution_friendly::(&a, &b, G2); let z = convolution_friendly::(&a, &b, G3); let mut mr = [(0, 0); 3]; for i in 0..d { mr[0] = (P1::m(), x[i]); mr[1] = (P2::m(), y[i]); mr[2] = (P3::m(), z[i]); ret[i] = garner(mr.to_vec(), mo); } } pub fn arbmod_convolution_modint( a: &[ModInt

], b: &[ModInt

], ret: &mut [ModInt

]) { let mo = P::m(); unsafe { arbmod_convolution(std::mem::transmute(a), std::mem::transmute(b), mo, std::mem::transmute(ret)); } } } // Verified by: yukicoder No.1112 // https://yukicoder.me/submissions/510746 // https://en.wikipedia.org/wiki/Berlekamp%E2%80%93Massey_algorithm // Complexity: O(n^2) // Depends on MInt.rs fn berlekamp_massey( n: usize, s: &[mod_int::ModInt

], ) -> Vec>{ type ModInt

= mod_int::ModInt

; let mut b = ModInt::new(1); let mut cp = vec![ModInt::new(0); n + 1]; let mut bp = vec![mod_int::ModInt::new(0); n]; cp[0] = mod_int::ModInt::new(1); bp[0] = mod_int::ModInt::new(1); let mut m = 1; let mut l = 0; for i in 0..2 * n + 1 { assert!(i >= l); assert!(l <= n); if i == 2 * n { break; } let mut d = s[i]; for j in 1..l + 1 { d += cp[j] * s[i - j]; } if d == ModInt::new(0) { m += 1; continue; } if 2 * l > i { // cp -= d/b * x^m * bp let factor = d * b.inv(); for j in 0..n + 1 - m { cp[m + j] -= factor * bp[j]; } m += 1; continue; } let factor = d * b.inv(); let tp = cp.clone(); for j in 0..n + 1 - m { cp[m + j] -= factor * bp[j]; } bp = tp; b = d; l = i + 1 - l; m = 1; } cp[0..l + 1].to_vec() } fn convolution(a: &[MInt], b: &[MInt]) -> Vec { if a.is_empty() || b.is_empty() { return vec![]; } let n = a.len() - 1; let m = b.len() - 1; let mut p = 1; while p < n + m + 1 { p *= 2; } let mut ans = vec![MInt::new(0); p]; let mut a = a.to_vec(); let mut b = b.to_vec(); a.resize(p, 0.into()); b.resize(p, 0.into()); arbitrary_mod::arbmod_convolution_modint(&a, &b, &mut ans); ans.truncate(n + m + 1); ans } // Finds [x^n] p(x)/q(x) // Ref: https://qiita.com/ryuhe1/items/da5acbcce4ac1911f47a // Verified by: https://atcoder.jp/contests/tdpc/submissions/24583334 fn bostan_mori(p: &[MInt], q: &[MInt], mut n: i64) -> MInt { assert!(p.len() < q.len()); let mut p = p.to_vec(); let mut q = q.to_vec(); while n > 0 { let mut qn = q.clone(); for i in 0..qn.len() { if i % 2 == 1 { qn[i] = -qn[i]; } } let num = convolution(&p, &qn); let den = convolution(&q, &qn); let mut nxt_p = vec![MInt::new(0); q.len() - 1]; let mut nxt_q = vec![MInt::new(0); q.len()]; for i in 0..q.len() - 1 { let to = 2 * i + (n % 2) as usize; if to < num.len() { nxt_p[i] = num[to]; } } for i in 0..q.len() { nxt_q[i] = den[2 * i]; } p = nxt_p; q = nxt_q; n /= 2; } p[0] * q[0].inv() } // Finds u a^e v^T by using Berlekamp-massey algorithm. // The linear map a is given as a closure. // Complexity: O(n^2 log e + nT(n)) where n = |u| and T(n) = complexity of a. // Ref: https://yukicoder.me/wiki/black_box_linear_algebra fn eval_matpow Vec>(mut a: F, e: i64, u: &[MInt], v: &[MInt]) -> MInt { let k = u.len(); // Find first 2k terms let mut terms = vec![MInt::new(0); 2 * k]; let mut cur = u.to_vec(); for pos in 0..2 * k { for i in 0..k { terms[pos] += cur[i] * v[i]; } cur = a(&cur); } let poly = berlekamp_massey(k, &terms); let mut nom = convolution(&terms[..k], &poly); nom.truncate(k); bostan_mori(&nom, &poly, e) } fn get_trans(a: [usize; 6], c: usize) -> Vec { let len = a[5] * c + 1; let mut dp = vec![vec![MInt::new(0); len]; c + 1]; dp[0][0] += 1; for &v in &a { // *= (1-x^{v{p+1}}y^{p+1}) / (1 - x^vy) for j in 0..c { for i in 0..len - v { dp[j + 1][i + v] = dp[j + 1][i + v] + dp[j][i]; } } } dp[c].to_vec() } // https://yukicoder.me/problems/no/215 (6) // 行列累乗でやろうとすると 7500^3 回の計算を要するため、kitamasa 法を使う。数列のゼロ化多項式がわかれば、最初の 7500 項程度を計算することで Bostan-Mori が使えて O(7500^2 log N)。 // 数列のゼロ化多項式は Berlekamp-Massey で O(7500^2) 程度で計算できるはずなので、これで計算できる。 // -> Bostan-Mori ではなく kitamasa 法を使って TLE。 fn main() { let n: i64 = get(); let p: usize = get(); let c: usize = get(); let len = p * 13 + c * 12 + 1; let mut trans = vec![MInt::new(0); len]; trans[0] += 1; let ps = [2, 3, 5, 7, 11, 13]; let cs = [4, 6, 8, 9, 10, 12]; let ptrans = get_trans(ps, p); let ctrans = get_trans(cs, c); for i in 0..ptrans.len() { for j in 0..ctrans.len() { trans[i + j] += ptrans[i] * ctrans[j]; } } let a = |u: &[MInt]| { let mut v = vec![MInt::new(0); len - 1]; for i in 0..len - 2 { v[i + 1] = u[i]; } for i in 0..len - 1 { v[0] += u[i] * trans[i + 1]; } v }; let mut start = vec![MInt::new(0); len - 1]; start[0] += 1; let mut rec = vec![MInt::new(0); len - 1]; for i in (0..len - 1).rev() { rec[i] = trans[i + 1]; if i + 1 < len - 1 { rec[i] = rec[i + 1] + rec[i]; } } let val = eval_matpow(a, n - 1, &start, &rec); println!("{}", val); }