// exp のmod 3 で1の項のM乗のN項目が答え // exp でmod3 で 1の項を取り出すには? // 3乗根をw と置いて // // f(x) + f(wx) + f(w^2x) で0が取り出せる // f(x) + w^2 f(wx) + wf(w^2x) でOKなはず // // N! [x^n] (e^x + w^2 e(wx) + w e^(w^2x))^m // = N! [x^n] sum_{0 <= i, j, i + j <= m} M!/i!j!(M-i-j)! * w^(2j + M-i-j) * e^(ix + jwx + (M-i-j)w^2x) // = sum_{i, j} C_{i, j} * w^(M-i+j) * (i+jw+(M-i-j)w^2)^n type M = ModInt<998_244_353>; fn main() { input!(n: usize, m: usize); let mut ans = M::zero(); let pc = Precalc::new(m); let w = P(M::zero(), M::one()); for i in 0..=m { for j in 0..=(m - i) { let k = m - i - j; let mut r = P(M::from(i), M::zero()) + P(M::from(j), M::zero()) * w + P(M::from(k), M::zero()) * w * w; let mut n = n; let mut t = P(M::one(), M::zero()); while n > 0 { if n & 1 == 1 { t = t * r; } r = r * r; n >>= 1; } for _ in 0..((2 * j + k) % 3) { t = t * w; } let val = t.0 * pc.fact(m) * pc.ifact(i) * pc.ifact(j) * pc.ifact(k); ans += val; } } ans *= M::new(3).inv().pow(m as u64); println!("{}", ans); } #[derive(Clone, Copy, Debug)] struct P(M, M); impl Add for P { type Output = Self; fn add(self, rhs: Self) -> Self { Self(self.0 + rhs.0, self.1 + rhs.1) } } impl Mul for P { type Output = Self; fn mul(self, rhs: Self) -> Self { let p = self.1 * rhs.1; Self(self.0 * rhs.0 - p, self.0 * rhs.1 + self.1 * rhs.0 - p) } } // ---------- begin input macro ---------- // reference: https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 #[macro_export] macro_rules! input { (source = $s:expr, $($r:tt)*) => { let mut iter = $s.split_whitespace(); input_inner!{iter, $($r)*} }; ($($r:tt)*) => { let s = { use std::io::Read; let mut s = String::new(); std::io::stdin().read_to_string(&mut s).unwrap(); s }; let mut iter = s.split_whitespace(); input_inner!{iter, $($r)*} }; } #[macro_export] macro_rules! input_inner { ($iter:expr) => {}; ($iter:expr, ) => {}; ($iter:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($iter, $t); input_inner!{$iter $($r)*} }; } #[macro_export] macro_rules! read_value { ($iter:expr, ( $($t:tt),* )) => { ( $(read_value!($iter, $t)),* ) }; ($iter:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($iter, $t)).collect::>() }; ($iter:expr, chars) => { read_value!($iter, String).chars().collect::>() }; ($iter:expr, bytes) => { read_value!($iter, String).bytes().collect::>() }; ($iter:expr, usize1) => { read_value!($iter, usize) - 1 }; ($iter:expr, $t:ty) => { $iter.next().unwrap().parse::<$t>().expect("Parse error") }; } // ---------- end input macro ---------- use std::ops::*; // ---------- begin trait ---------- pub trait Zero: Sized + Add { fn zero() -> Self; fn is_zero(&self) -> bool; } pub trait One: Sized + Mul { fn one() -> Self; fn is_one(&self) -> bool; } pub trait Ring: Zero + One + Sub {} pub trait Field: Ring + Div {} // ---------- end trait ---------- // ---------- begin modint ---------- pub const fn pow_mod(mut r: u32, mut n: u32, m: u32) -> u32 { let mut t = 1; while n > 0 { if n & 1 == 1 { t = (t as u64 * r as u64 % m as u64) as u32; } r = (r as u64 * r as u64 % m as u64) as u32; n >>= 1; } t } pub const fn primitive_root(p: u32) -> u32 { let mut m = p - 1; let mut f = [1; 30]; let mut k = 0; let mut d = 2; while d * d <= m { if m % d == 0 { f[k] = d; k += 1; } while m % d == 0 { m /= d; } d += 1; } if m > 1 { f[k] = m; k += 1; } let mut g = 1; while g < p { let mut ok = true; let mut i = 0; while i < k { ok &= pow_mod(g, (p - 1) / f[i], p) > 1; i += 1; } if ok { break; } g += 1; } g } pub const fn is_prime(n: u32) -> bool { if n <= 1 { return false; } let mut d = 2; while d * d <= n { if n % d == 0 { return false; } d += 1; } true } #[derive(Clone, Copy, PartialEq, Eq)] pub struct ModInt(u32); impl ModInt<{ M }> { const REM: u32 = { let mut t = 1u32; let mut s = !M + 1; let mut n = !0u32 >> 2; while n > 0 { if n & 1 == 1 { t = t.wrapping_mul(s); } s = s.wrapping_mul(s); n >>= 1; } t }; const INI: u64 = ((1u128 << 64) % M as u128) as u64; const IS_PRIME: () = assert!(is_prime(M)); const PRIMITIVE_ROOT: u32 = primitive_root(M); const ORDER: usize = 1 << (M - 1).trailing_zeros(); const fn reduce(x: u64) -> u32 { let _ = Self::IS_PRIME; let b = (x as u32 * Self::REM) as u64; let t = x + b * M as u64; let mut c = (t >> 32) as u32; if c >= M { c -= M; } c as u32 } const fn multiply(a: u32, b: u32) -> u32 { Self::reduce(a as u64 * b as u64) } pub const fn new(v: u32) -> Self { assert!(v < M); Self(Self::reduce(v as u64 * Self::INI)) } pub const fn const_mul(&self, rhs: Self) -> Self { Self(Self::multiply(self.0, rhs.0)) } pub const fn pow(&self, mut n: u64) -> Self { let mut t = Self::new(1); let mut r = *self; while n > 0 { if n & 1 == 1 { t = t.const_mul(r); } r = r.const_mul(r); n >>= 1; } t } pub const fn inv(&self) -> Self { assert!(self.0 != 0); self.pow(M as u64 - 2) } pub const fn get(&self) -> u32 { Self::reduce(self.0 as u64) } pub const fn zero() -> Self { Self::new(0) } pub const fn one() -> Self { Self::new(1) } } impl Add for ModInt<{ M }> { type Output = Self; fn add(self, rhs: Self) -> Self::Output { let mut v = self.0 + rhs.0; if v >= M { v -= M; } Self(v) } } impl Sub for ModInt<{ M }> { type Output = Self; fn sub(self, rhs: Self) -> Self::Output { let mut v = self.0 - rhs.0; if self.0 < rhs.0 { v += M; } Self(v) } } impl Mul for ModInt<{ M }> { type Output = Self; fn mul(self, rhs: Self) -> Self::Output { self.const_mul(rhs) } } impl Div for ModInt<{ M }> { type Output = Self; fn div(self, rhs: Self) -> Self::Output { self * rhs.inv() } } impl AddAssign for ModInt<{ M }> { fn add_assign(&mut self, rhs: Self) { *self = *self + rhs; } } impl SubAssign for ModInt<{ M }> { fn sub_assign(&mut self, rhs: Self) { *self = *self - rhs; } } impl MulAssign for ModInt<{ M }> { fn mul_assign(&mut self, rhs: Self) { *self = *self * rhs; } } impl DivAssign for ModInt<{ M }> { fn div_assign(&mut self, rhs: Self) { *self = *self / rhs; } } impl Neg for ModInt<{ M }> { type Output = Self; fn neg(self) -> Self::Output { if self.0 == 0 { self } else { Self(M - self.0) } } } impl std::fmt::Display for ModInt<{ M }> { fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result { write!(f, "{}", self.get()) } } impl std::fmt::Debug for ModInt<{ M }> { fn fmt<'a>(&self, f: &mut std::fmt::Formatter<'a>) -> std::fmt::Result { write!(f, "{}", self.get()) } } impl std::str::FromStr for ModInt<{ M }> { type Err = std::num::ParseIntError; fn from_str(s: &str) -> Result { let val = s.parse::()?; Ok(ModInt::new(val)) } } impl From for ModInt<{ M }> { fn from(val: usize) -> ModInt<{ M }> { ModInt::new((val % M as usize) as u32) } } // ---------- end modint ---------- // ---------- begin precalc ---------- pub struct Precalc { fact: Vec>, ifact: Vec>, inv: Vec>, } impl Precalc { pub fn new(size: usize) -> Self { let mut fact = vec![ModInt::one(); size + 1]; let mut ifact = vec![ModInt::one(); size + 1]; let mut inv = vec![ModInt::one(); size + 1]; for i in 2..=size { fact[i] = fact[i - 1] * ModInt::from(i); } ifact[size] = fact[size].inv(); for i in (2..=size).rev() { inv[i] = ifact[i] * fact[i - 1]; ifact[i - 1] = ifact[i] * ModInt::from(i); } Self { fact, ifact, inv } } pub fn fact(&self, n: usize) -> ModInt { self.fact[n] } pub fn ifact(&self, n: usize) -> ModInt { self.ifact[n] } pub fn inv(&self, n: usize) -> ModInt { assert!(0 < n); self.inv[n] } pub fn perm(&self, n: usize, k: usize) -> ModInt { if k > n { return ModInt::zero(); } self.fact[n] * self.ifact[n - k] } pub fn binom(&self, n: usize, k: usize) -> ModInt { if n < k { return ModInt::zero(); } self.fact[n] * self.ifact[k] * self.ifact[n - k] } } // ---------- end precalc ---------- impl Zero for ModInt<{ M }> { fn zero() -> Self { Self::zero() } fn is_zero(&self) -> bool { self.0 == 0 } } impl One for ModInt<{ M }> { fn one() -> Self { Self::one() } fn is_one(&self) -> bool { self.get() == 1 } } impl Ring for ModInt<{ M }> {} impl Field for ModInt<{ M }> {} // ---------- begin array op ----------