#[allow(unused_imports)] use std::cmp::*; #[allow(unused_imports)] use std::collections::*; 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; } } } #[allow(dead_code)] 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 = 998_244_353; 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; } } } } // Depends on: fft.rs, MInt.rs // Verified by: ABC269-Ex (https://atcoder.jp/contests/abc269/submissions/39116328) pub struct FPSOps { gen: mod_int::ModInt, } impl FPSOps { pub fn new(gen: mod_int::ModInt) -> Self { FPSOps { gen: gen } } } impl FPSOps { pub fn add(&self, mut a: Vec>, mut b: Vec>) -> Vec> { if a.len() < b.len() { std::mem::swap(&mut a, &mut b); } for i in 0..b.len() { a[i] += b[i]; } a } pub fn mul(&self, a: Vec>, b: Vec>) -> Vec> { type MInt = mod_int::ModInt; 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 { p *= 2; } let mut f = vec![MInt::new(0); p]; let mut g = vec![MInt::new(0); p]; for i in 0..n + 1 { f[i] = a[i]; } for i in 0..m + 1 { g[i] = b[i]; } let fac = MInt::new(p as i64).inv(); let zeta = self.gen.pow((M::m() - 1) / p as i64); fft::fft(&mut f, zeta, 1.into()); fft::fft(&mut g, zeta, 1.into()); for i in 0..p { f[i] *= g[i] * fac; } fft::inv_fft(&mut f, zeta.inv(), 1.into()); f.truncate(n + m + 1); f } } // Finds [x^n] p(x)/q(x) // Ref: https://qiita.com/ryuhe1/items/da5acbcce4ac1911f47a // Verified by: https://atcoder.jp/contests/tdpc/submissions/24583334 // Depends on: MInt.rs fn bostan_mori(ops: &FPSOps

, p: &[MInt], q: &[MInt], mut n: i64) -> MInt { if p.is_empty() { return 0.into(); } 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 = ops.mul(p, qn.clone()); let den = ops.mul(q.clone(), 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() } fn main() { let n: i64 = get(); let m: usize = get(); let ops = FPSOps::new(MInt::new(3)); let mut num = vec![]; let mut den = vec![MInt::new(1)]; for i in 2..m + 1 { // g += (x+...+x^{i-1}) / (1+...+x^{i-1}-x^i) let mut num1 = vec![MInt::new(1); i]; num1[0] -= 1; let mut den1 = vec![MInt::new(1); i + 1]; den1[i] -= 2; let newnum = ops.mul(num.clone(), den1.clone()); let newnum = ops.add(newnum, ops.mul(den.clone(), num1.clone())); let newden = ops.mul(den.clone(), den1); num = newnum; den = newden; } // g / (1 - g) = num / (den - num) for i in 0..num.len() { den[i] -= num[i]; } let ans = bostan_mori(&ops, &num, &den, n); println!("{}", ans); }