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 = 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; } } } } // Computes f^{-1} mod x^{f.len()}. // Reference: https://codeforces.com/blog/entry/56422 // Complexity: O(n log n) // Verified by: https://judge.yosupo.jp/submission/3219 // Depends on: MInt.rs, fft.rs fn fps_inv( f: &[mod_int::ModInt

], gen: mod_int::ModInt

) -> Vec> { let n = f.len(); assert!(n.is_power_of_two()); assert_eq!(f[0], 1.into()); let mut sz = 1; let mut r = vec![mod_int::ModInt::new(0); n]; let mut tmp_f = vec![mod_int::ModInt::new(0); n]; let mut tmp_r = vec![mod_int::ModInt::new(0); n]; r[0] = 1.into(); // Adopts the technique used in https://judge.yosupo.jp/submission/3153 while sz < n { let zeta = gen.pow((P::m() - 1) / sz as i64 / 2); tmp_f[..2 * sz].copy_from_slice(&f[..2 * sz]); tmp_r[..2 * sz].copy_from_slice(&r[..2 * sz]); fft::fft(&mut tmp_r[..2 * sz], zeta, 1.into()); fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into()); let fac = mod_int::ModInt::new(2 * sz as i64).inv().pow(2); for i in 0..2 * sz { tmp_f[i] *= tmp_r[i]; } fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into()); for v in &mut tmp_f[..sz] { *v = 0.into(); } fft::fft(&mut tmp_f[..2 * sz], zeta, 1.into()); for i in 0..2 * sz { tmp_f[i] = -tmp_f[i] * tmp_r[i] * fac; } fft::inv_fft(&mut tmp_f[..2 * sz], zeta.inv(), 1.into()); r[sz..2 * sz].copy_from_slice(&tmp_f[sz..2 * sz]); sz *= 2; } r } // Depends on MInt.rs fn fact_init(w: usize) -> (Vec, Vec) { let mut fac = vec![MInt::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) } // Computes exp(f) mod x^{f.len()}. // Reference: https://arxiv.org/pdf/1301.5804.pdf // Complexity: O(n log n) // Depends on: MInt.rs, fact_init.rs, fft.rs fn fps_exp( h: &[mod_int::ModInt

], gen: mod_int::ModInt

, fac: &[mod_int::ModInt

], invfac: &[mod_int::ModInt

], ) -> Vec> { let n = h.len(); assert!(n.is_power_of_two()); assert_eq!(h[0], 0.into()); let mut m = 1; let mut f = vec![mod_int::ModInt::new(0); n]; let mut g = vec![mod_int::ModInt::new(0); n]; let mut tmp_f = vec![mod_int::ModInt::new(0); n]; let mut tmp_g = vec![mod_int::ModInt::new(0); n]; let mut tmp = vec![mod_int::ModInt::new(0); n]; f[0] = 1.into(); g[0] = 1.into(); // Adopts the technique used in https://judge.yosupo.jp/submission/3153 while m < n { // upheld invariants: f = exp(h) (mod x^m) // g = exp(-h) (mod x^(m/2)) // Complexity: 4 * fft(2 * m) + 2 * fft(m) + 2 * inv_fft(2 * m) + 3 * inv_fft(m) // ~= 8.5 * fft(2 * m) let zeta2m = gen.pow((P::m() - 1) / m as i64 / 2); let zeta = zeta2m * zeta2m; // 2.a': g = 2g - fg^2 mod x^m let factor2m = mod_int::ModInt::new(m as i64 * 2).inv(); let factor = factor2m * 2; let factor2 = factor * factor; // Here we only need FFT(f[..m]), but we use it later at 2.c' tmp_f[..2 * m].copy_from_slice(&f[..2 * m]); fft::fft(&mut tmp_f[..2 * m], zeta2m, 1.into()); if m > 1 { // The following can be dropped because the actual // computation was done in the previous iteration. // tmp_g[..m].copy_from_slice(&g[..m]); // fft::fft(&mut tmp_g[..m], zeta, 1.into()); for i in 0..m { tmp[i] = tmp_f[i] * tmp_g[i]; } fft::inv_fft(&mut tmp[..m], zeta.inv(), 1.into()); for v in &mut tmp[..m / 2] { *v = 0.into(); } fft::fft(&mut tmp[..m], zeta, 1.into()); for i in 0..m { tmp[i] = -tmp[i] * tmp_g[i] * factor2; } fft::inv_fft(&mut tmp[..m], zeta.inv(), 1.into()); g[m / 2..m].copy_from_slice(&tmp[m / 2..m]); } // 2.b': q = h' mod x^(m-1) for i in 0..m - 1 { tmp[i] = h[i + 1] * (i + 1) as i64; } tmp[m - 1] = 0.into(); // 2.c': r = fq (mod x^m - 1) fft::fft(&mut tmp[..m], zeta, 1.into()); // FFT(f[..2m])[..m] == FFT(f[..m]) // Note that the result of FFT is bit-reversed. for i in 0..m { tmp[i] *= tmp_f[i] * factor; } fft::inv_fft(&mut tmp[..m], zeta.inv(), 1.into()); // 2.d' s = x(f' - r) mod (x^m - 1) for i in (0..m - 1).rev() { tmp.swap(i, i + 1); } for i in 0..m { tmp[i] = f[i] * i as i64 - tmp[i]; } // 2.e': t = gs mod x^m tmp_g[..2 * m].copy_from_slice(&g[..2 * m]); fft::fft(&mut tmp_g[..2 * m], zeta2m, 1.into()); fft::fft(&mut tmp[..2 * m], zeta2m, 1.into()); for i in 0..2 * m { tmp[i] *= tmp_g[i] * factor2m; } fft::inv_fft(&mut tmp[..2 * m], zeta2m.inv(), 1.into()); // 2.f': u = (h mod x^2m - \int tx^(m-1)) / x^m for i in 0..m { tmp[i] = h[i + m] - tmp[i] * fac[i + m - 1] * invfac[i + m]; } for v in &mut tmp[m..2 * m] { *v = 0.into(); } // 2.g': v = fu mod x^m fft::fft(&mut tmp[..2 * m], zeta2m, 1.into()); for i in 0..2 * m { tmp[i] *= tmp_f[i] * factor2m; } fft::inv_fft(&mut tmp[..2 * m], zeta2m.inv(), 1.into()); // 2.h': f += vx^m f[m..2 * m].copy_from_slice(&tmp[..m]); // 2.i': m *= 2 m *= 2; } f } // Computes ln f mod x^{f.len()}. // Reference: https://codeforces.com/blog/entry/56422 // Complexity: O(n log n) // Verified by: https://judge.yosupo.jp/submission/53708 // Depends on: MInt.rs, fact_init.rs, fft.rs, fps/fps_inv.rs fn fps_ln( f: &[mod_int::ModInt

], gen: mod_int::ModInt

, fac: &[mod_int::ModInt

], invfac: &[mod_int::ModInt

], ) -> Vec> { let n = f.len(); assert!(n.is_power_of_two()); assert_eq!(f[0], 1.into()); let mut inv = fps_inv(&f, gen); let mut der = vec![mod_int::ModInt::new(0); 2 * n]; for i in 1..n { der[i - 1] = f[i] * i as i64; } inv.resize(2 * n, 0.into()); let zeta = gen.pow((P::m() - 1) / n as i64 / 2); fft::fft(&mut der, zeta, 1.into()); fft::fft(&mut inv, zeta, 1.into()); let invlen = mod_int::ModInt::new(2 * n as i64).inv(); for i in 0..2 * n { der[i] *= inv[i] * invlen; } fft::inv_fft(&mut der, zeta.inv(), 1.into()); // integral of f'/f let mut ans = vec![mod_int::ModInt::new(0); n]; for i in 1..n { ans[i] = der[i - 1] * invfac[i] * fac[i - 1]; } ans } // https://yukicoder.me/problems/no/2413 (4) // A = 1 + x + ... + x^9, B = 1 + x^10 + ... + x^90 とする。 // 問題の答えは \sum_{0 <= i,j <= 9N, 99|i+10j}(i+j)^K[x^i]A^{ceil(N/2)}[x^j]A^{floor(N/2)} である。 // -> 解説を見た。A^? を計算してから、次数 mod 11 で見て畳み込みをする。 // 計算量は O(11 K log K) where K = 9N である。 // The author read the editorial before implementing this. fn main() { let n: usize = get(); let k: usize = get(); const W: usize = 1 << 20; let (fac, invfac) = fact_init(W + 1); let mut f = vec![MInt::new(0); W]; for i in 0..10 { f[i] += 1; } let mut g1 = fps_ln(&f, 3.into(), &fac, &invfac); let mut g2 = g1.clone(); for i in 0..W { g1[i] *= (n as i64 + 1) / 2; g2[i] *= n as i64 / 2; } let h1 = fps_exp(&g1, 3.into(), &fac, &invfac); let h2 = fps_exp(&g2, 3.into(), &fac, &invfac); let factor = MInt::new(W as i64).inv(); let zeta = MInt::new(3).pow((MOD - 1) / W as i64); let mut ans = MInt::new(0); let mut c1 = vec![MInt::new(0); W]; let mut c2 = vec![MInt::new(0); W]; for rem in 0..11 { for i in 0..W { if i % 11 == rem { c1[i] = h1[i]; c2[i] = h2[i]; } else { c1[i] = 0.into(); c2[i] = 0.into(); } } fft::fft(&mut c1, zeta, 1.into()); fft::fft(&mut c2, zeta, 1.into()); for i in 0..W { c1[i] *= c2[i] * factor; } fft::inv_fft(&mut c1, zeta.inv(), 1.into()); for i in 0..W { if i % 9 == 0 { ans += MInt::new(i as i64).pow(k as i64) * c1[i]; } } } println!("{}", ans); }