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ps/frac.hpp
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- Last update: 2024-11-09 00:49:07+00:00
- Include:
#include "ps/frac.hpp"
Depends on
- arith/sat.hpp
- bit/ctz.hpp
- mod/inv.hpp
- mod/modint.hpp
- mod/sqrt.hpp
prelude.hpp- ps/fft.hpp
- ps/fps.hpp
- ps/inv.hpp
- types.hpp
- util/rand.hpp
- util/seed.hpp
Required by
Verified with
Code
#pragma once
#include "ps/fps.hpp"
template <class T>
class fraction {
public:
using fps = formal_power_series<T>;
fps numer, denom;
fraction() : numer(fps::zero()), denom(fps::one()) { }
fraction(fps numer) : numer(numer), denom(fps::one()) { }
fraction(fps numer, fps denom) : numer(numer), denom(denom) { }
T eval(T x) const { return numer.eval(x) / denom.eval(x); }
fraction add(fraction rhs) && {
numer = move(numer).conv(rhs.denom).add(move(rhs.numer).conv(denom));
denom = move(denom).conv(move(rhs.denom));
return move(*this);
}
fraction add(fraction rhs) const& { return fraction(*this).add(move(rhs)); }
fraction sub(fraction rhs) && {
numer = move(numer).conv(rhs.denom).sub(move(rhs.numer).conv(denom));
denom = move(denom).conv(move(rhs.denom));
return move(*this);
}
fraction sub(fraction rhs) const& { return fraction(*this).sub(move(rhs)); }
fraction mul(fraction rhs) && {
numer = move(numer).conv(move(rhs.numer));
denom = move(denom).conv(move(rhs.denom));
return move(*this);
}
fraction mul(fraction rhs) const& { return fraction(*this).mul(move(rhs)); }
fraction div(fraction rhs) && {
numer = move(numer).conv(move(rhs.denom));
denom = move(denom).conv(move(rhs.numer));
return move(*this);
}
fraction div(fraction rhs) const& { return fraction(*this).div(move(rhs)); }
fraction diff() && {
fps pd = numer.diff(), qd = denom.diff();
numer = move(pd).conv(denom).sub(move(numer).conv(move(qd)));
denom = move(denom).square();
return move(*this);
}
fraction diff() const& { return fraction(*this).diff(); }
};
#line 1 "mod/modint.hpp"
#include <cassert>
#include <numeric>
#include <type_traits>
#ifdef _MSC_VER
#include <intrin.h>
#endif
#line 1 "atcoder/internal_math.hpp"
#include <utility>
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
return x;
}
// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
unsigned int _m;
unsigned long long im;
// @param m `1 <= m < 2^31`
explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
// @return m
unsigned int umod() const { return _m; }
// @param a `0 <= a < m`
// @param b `0 <= b < m`
// @return `a * b % m`
unsigned int mul(unsigned int a, unsigned int b) const {
// [1] m = 1
// a = b = im = 0, so okay
// [2] m >= 2
// im = ceil(2^64 / m)
// -> im * m = 2^64 + r (0 <= r < m)
// let z = a*b = c*m + d (0 <= c, d < m)
// a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
// c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
// ((ab * im) >> 64) == c or c + 1
unsigned long long z = a;
z *= b;
#ifdef _MSC_VER
unsigned long long x;
_umul128(z, im, &x);
#else
unsigned long long x =
(unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
unsigned int v = (unsigned int)(z - x * _m);
if (_m <= v) v += _m;
return v;
}
};
// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = safe_mod(x, m);
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);
// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return {b, 0};
// Contracts:
// [1] s - m0 * a = 0 (mod b)
// [2] t - m1 * a = 0 (mod b)
// [3] s * |m1| + t * |m0| <= b
long long s = b, t = a;
long long m0 = 0, m1 = 1;
while (t) {
long long u = s / t;
s -= t * u;
m0 -= m1 * u; // |m1 * u| <= |m1| * s <= b
// [3]:
// (s - t * u) * |m1| + t * |m0 - m1 * u|
// <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
// = s * |m1| + t * |m0| <= b
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
// by [3]: |m0| <= b/g
// by g != b: |m0| < b/g
if (m0 < 0) m0 += b / s;
return {s, m0};
}
// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
if (m == 2) return 1;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (int i = 3; (long long)(i)*i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) {
x /= i;
}
}
}
if (x > 1) {
divs[cnt++] = x;
}
for (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);
// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
unsigned long long m,
unsigned long long a,
unsigned long long b) {
unsigned long long ans = 0;
while (true) {
if (a >= m) {
ans += n * (n - 1) / 2 * (a / m);
a %= m;
}
if (b >= m) {
ans += n * (b / m);
b %= m;
}
unsigned long long y_max = a * n + b;
if (y_max < m) break;
// y_max < m * (n + 1)
// floor(y_max / m) <= n
n = (unsigned long long)(y_max / m);
b = (unsigned long long)(y_max % m);
std::swap(m, a);
}
return ans;
}
} // namespace internal
} // namespace atcoder
#line 1 "atcoder/internal_type_traits.hpp"
#line 7 "atcoder/internal_type_traits.hpp"
namespace atcoder {
namespace internal {
#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value ||
std::is_same<T, __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int128 =
typename std::conditional<std::is_same<T, __uint128_t>::value ||
std::is_same<T, unsigned __int128>::value,
std::true_type,
std::false_type>::type;
template <class T>
using make_unsigned_int128 =
typename std::conditional<std::is_same<T, __int128_t>::value,
__uint128_t,
unsigned __int128>;
template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
is_signed_int128<T>::value ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
std::is_signed<T>::value) ||
is_signed_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<(is_integral<T>::value &&
std::is_unsigned<T>::value) ||
is_unsigned_int128<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<
is_signed_int128<T>::value,
make_unsigned_int128<T>,
typename std::conditional<std::is_signed<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type>::type;
#else
template <class T> using is_integral = typename std::is_integral<T>;
template <class T>
using is_signed_int =
typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using is_unsigned_int =
typename std::conditional<is_integral<T>::value &&
std::is_unsigned<T>::value,
std::true_type,
std::false_type>::type;
template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
std::make_unsigned<T>,
std::common_type<T>>::type;
#endif
template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;
template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;
template <class T> using to_unsigned_t = typename to_unsigned<T>::type;
} // namespace internal
} // namespace atcoder
#line 14 "mod/modint.hpp"
namespace atcoder {
namespace internal {
struct modint_base {}; struct static_modint_base : modint_base {}; template <class T> using is_modint = std::is_base_of<modint_base, T>; template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;
} // namespace internal
template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
using mint = static_modint; public: static constexpr int mod() { return m; } static constexpr mint raw(int v) { mint x; x._v = v; return x; } constexpr static_modint() : _v(0) {} template <class T, internal::is_signed_int_t<T>* = nullptr> constexpr static_modint(T v) : _v() { long long x = (long long)(v % (long long)(umod())); if (x < 0) x += umod(); _v = (unsigned int)(x); } template <class T, internal::is_unsigned_int_t<T>* = nullptr> constexpr static_modint(T v) : _v() { _v = (unsigned int)(v % umod()); } constexpr unsigned int val() const { return _v; } constexpr mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } constexpr mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } constexpr mint operator++(int) { mint result = *this; ++*this; return result; } constexpr mint operator--(int) { mint result = *this; --*this; return result; } constexpr mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } constexpr mint& operator-=(const mint& rhs) { _v -= rhs._v; if (_v >= umod()) _v += umod(); return *this; } constexpr mint& operator*=(const mint& rhs) { unsigned long long z = _v; z *= rhs._v; _v = (unsigned int)(z % umod()); return *this; } constexpr mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } constexpr mint operator+() const { return *this; } constexpr mint operator-() const { return mint() - *this; } constexpr mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } constexpr mint inv() const { if (prime) { assert(_v); return pow(umod() - 2); } else { auto eg = internal::inv_gcd(_v, m); assert(eg.first == 1); return eg.second; } } constexpr friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } constexpr friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } constexpr friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } constexpr friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } constexpr friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } constexpr friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } constexpr friend bool operator<(const mint& lhs, const mint& rhs) { return lhs._v < rhs._v; } private: unsigned int _v; static constexpr unsigned int umod() { return m; } static constexpr bool prime = internal::is_prime<m>;
};
template <int id> struct dynamic_modint : internal::modint_base {
using mint = dynamic_modint; public: static int mod() { return (int)(bt.umod()); } static void set_mod(int m) { assert(1 <= m); bt = internal::barrett(m); } static mint raw(int v) { mint x; x._v = v; return x; } dynamic_modint() : _v(0) {} template <class T, internal::is_signed_int_t<T>* = nullptr> dynamic_modint(T v) { long long x = (long long)(v % (long long)(mod())); if (x < 0) x += mod(); _v = (unsigned int)(x); } template <class T, internal::is_unsigned_int_t<T>* = nullptr> dynamic_modint(T v) { _v = (unsigned int)(v % mod()); } unsigned int val() const { return _v; } mint& operator++() { _v++; if (_v == umod()) _v = 0; return *this; } mint& operator--() { if (_v == 0) _v = umod(); _v--; return *this; } mint operator++(int) { mint result = *this; ++*this; return result; } mint operator--(int) { mint result = *this; --*this; return result; } mint& operator+=(const mint& rhs) { _v += rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator-=(const mint& rhs) { _v += mod() - rhs._v; if (_v >= umod()) _v -= umod(); return *this; } mint& operator*=(const mint& rhs) { _v = bt.mul(_v, rhs._v); return *this; } mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); } mint operator+() const { return *this; } mint operator-() const { return mint() - *this; } mint pow(long long n) const { assert(0 <= n); mint x = *this, r = 1; while (n) { if (n & 1) r *= x; x *= x; n >>= 1; } return r; } mint inv() const { auto eg = internal::inv_gcd(_v, mod()); assert(eg.first == 1); return eg.second; } friend mint operator+(const mint& lhs, const mint& rhs) { return mint(lhs) += rhs; } friend mint operator-(const mint& lhs, const mint& rhs) { return mint(lhs) -= rhs; } friend mint operator*(const mint& lhs, const mint& rhs) { return mint(lhs) *= rhs; } friend mint operator/(const mint& lhs, const mint& rhs) { return mint(lhs) /= rhs; } friend bool operator==(const mint& lhs, const mint& rhs) { return lhs._v == rhs._v; } friend bool operator!=(const mint& lhs, const mint& rhs) { return lhs._v != rhs._v; } friend bool operator<(const mint& lhs, const mint& rhs) { return lhs._v < rhs._v; } private: unsigned int _v; static internal::barrett bt; static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
namespace internal {
template <class T> using is_static_modint = std::is_base_of<internal::static_modint_base, T>; template <class T> using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>; template <class> struct is_dynamic_modint : public std::false_type {}; template <int id> struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {}; template <class T> using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;
} // namespace internal
} // namespace atcoder
#line 2 "prelude.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using vi = vector<int>;
using vvi = vector<vector<int>>;
using vll = vector<ll>;
using vvll = vector<vector<ll>>;
using vc = vector<char>;
#define rep2(i, m, n) for (auto i = (m); i < (n); i++)
#define rep(i, n) rep2(i, 0, n)
#define repr2(i, m, n) for (auto i = (n); i-- > (m);)
#define repr(i, n) repr2(i, 0, n)
#define all(x) begin(x), end(x)
auto ndvec(int n, auto e) { return vector(n, e); }
auto ndvec(int n, auto ...e) { return vector(n, ndvec(e...)); }
auto comp_key(auto&& f) { return [&](auto&& a, auto&& b) { return f(a) < f(b); }; }
auto& max(const auto& a, const auto& b) { return a < b ? b : a; }
auto& min(const auto& a, const auto& b) { return b < a ? b : a; }
#if __cpp_lib_ranges
namespace R = std::ranges;
namespace V = std::views;
#endif
#line 4 "mod/inv.hpp"
// #include <atcoder/modint>
template <class T = atcoder::modint998244353>
T inverse(int n) {
static vector<T> v = {T(0), T(1)};
while (v.size() <= n) {
v.push_back(-v[T::mod() % v.size()] * (T::mod() / v.size()));
}
return v[n];
}
#line 3 "bit/ctz.hpp"
#pragma GCC target("bmi")
template <class T>
int ctz(T x) {
if (!x) return sizeof(T) * 8;
if constexpr (sizeof(T) <= sizeof(unsigned)) {
return __builtin_ctz((unsigned)x);
} else if constexpr (sizeof(T) <= sizeof(unsigned long long)) {
return __builtin_ctzll((unsigned long long)x);
} else if constexpr (sizeof(T) <= sizeof(unsigned long long) * 2) {
unsigned long long y = x;
return y ? ctz(y)
: sizeof(y) * 8 + ctz((unsigned long long)(x >> sizeof(y) * 8));
}
}
#line 3 "util/seed.hpp"
auto seed() {
#if defined(LOCAL) && !defined(NO_FIX_SEED)
return 314169265258979;
#endif
return chrono::steady_clock::now().time_since_epoch().count();
}
#line 3 "util/rand.hpp"
uint32_t rand32() {
static uint32_t x = seed();
x ^= x << 13;
x ^= x >> 17;
x ^= x << 5;
return x;
}
uint64_t rand64() {
return uint64_t(rand32()) << 32 | rand32();
}
#line 4 "mod/sqrt.hpp"
template <class T>
optional<T> mod_sqrt(T a) {
// Tonelli-Shanks
if (T::mod() <= 2) return a;
if (a == T(0)) return T(0);
if (a.pow((T::mod() - 1) / 2) == -1) return nullopt;
int s = ctz(T::mod() - 1);
int q = (T::mod() - 1) >> s;
T x = a.pow((q + 1) / 2);
T b = rand32();
while (b.pow((T::mod() - 1) / 2) != -1) b = rand32();
b = b.pow(q);
T ia = a.inv();
s -= 2;
for (T e = ia * x * x; e != 1; b *= b, s--) {
if (e.pow(1 << s) != 1) x *= b, e = ia * x * x;
}
return x;
}
#line 3 "arith/sat.hpp"
template <class T, class U>
auto sat_add(T a, U b) {
using V = common_type_t<T, U>;
V res;
return __builtin_add_overflow((V)a, (V)b, &res)
? (a < 0 ? numeric_limits<V>::min() : numeric_limits<V>::max())
: res;
}
template <class T, class U>
auto sat_sub(T a, U b) {
using V = common_type_t<T, U>;
V res;
return __builtin_sub_overflow((V)a, (V)b, &res)
? (a < 0 ? numeric_limits<V>::min() : numeric_limits<V>::max())
: res;
}
template <class T, class U>
auto sat_mul(T a, U b) {
using V = common_type_t<T, U>;
V res;
return __builtin_mul_overflow((V)a, (V)b, &res)
? ((a < 0) == (b < 0) ? numeric_limits<V>::max()
: numeric_limits<V>::min())
: res;
}
#line 3 "types.hpp"
template <class It>
using val_t = typename iterator_traits<It>::value_type;
#line 4 "ps/fft.hpp"
#line 1 "atcoder/internal_bit.hpp"
#ifdef _MSC_VER
#include <intrin.h>
#endif
namespace atcoder {
namespace internal {
// @param n `0 <= n`
// @return minimum non-negative `x` s.t. `n <= 2**x`
int ceil_pow2(int n) {
int x = 0;
while ((1U << x) < (unsigned int)(n)) x++;
return x;
}
// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
constexpr int bsf_constexpr(unsigned int n) {
int x = 0;
while (!(n & (1 << x))) x++;
return x;
}
// @param n `1 <= n`
// @return minimum non-negative `x` s.t. `(n & (1 << x)) != 0`
int bsf(unsigned int n) {
#ifdef _MSC_VER
unsigned long index;
_BitScanForward(&index, n);
return index;
#else
return __builtin_ctz(n);
#endif
}
} // namespace internal
} // namespace atcoder
#line 8 "ps/fft.hpp"
#ifndef ATCODER_CONVOLUTION_HPP
#define ATCODER_CONVOLUTION_HPP
namespace atcoder {
namespace internal {
template <
class mint, int g = internal::primitive_root<mint::mod()>,
internal::is_static_modint_t<mint>* = nullptr>
struct fft_info {
static constexpr int rank2 = bsf_constexpr(mint::mod() - 1);
mint root[rank2 + 1]; // root[i]^(2^i) == 1
mint iroot[rank2 + 1]; // root[i] * iroot[i] == 1
mint rate2[std::max(0, rank2 - 2 + 1)];
mint irate2[std::max(0, rank2 - 2 + 1)];
mint rate3[std::max(0, rank2 - 3 + 1)];
mint irate3[std::max(0, rank2 - 3 + 1)];
constexpr fft_info() : root(), iroot(), rate2(), irate2(), rate3(), irate3() {
root[rank2] = mint(g).pow((mint::mod() - 1) >> rank2);
iroot[rank2] = root[rank2].inv();
for (int i = rank2 - 1; i >= 0; i--) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
{
mint prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 2; i++) {
rate2[i] = root[i + 2] * prod;
irate2[i] = iroot[i + 2] * iprod;
prod *= iroot[i + 2];
iprod *= root[i + 2];
}
}
{
mint prod = 1, iprod = 1;
for (int i = 0; i <= rank2 - 3; i++) {
rate3[i] = root[i + 3] * prod;
irate3[i] = iroot[i + 3] * iprod;
prod *= iroot[i + 3];
iprod *= root[i + 3];
}
}
}
};
template <class It, internal::is_static_modint_t<val_t<It>>* = nullptr>
void butterfly(It a, It last) {
using mint = val_t<It>;
int n = last - a;
int h = internal::ceil_pow2(n);
static constexpr fft_info<mint> info;
int len = 0; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len < h) {
if (h - len == 1) {
int p = 1 << (h - len - 1);
mint rot = 1;
for (int s = 0; s < (1 << len); s++) {
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p] * rot;
a[i + offset] = l + r;
a[i + offset + p] = l - r;
}
if (s + 1 != (1 << len)) rot *= info.rate2[bsf(~(unsigned int)(s))];
}
len++;
} else {
// 4-base
int p = 1 << (h - len - 2);
mint rot = 1, imag = info.root[2];
for (int s = 0; s < (1 << len); s++) {
mint rot2 = rot * rot;
mint rot3 = rot2 * rot;
int offset = s << (h - len);
for (int i = 0; i < p; i++) {
auto mod2 = 1ULL * mint::mod() * mint::mod();
auto a0 = 1ULL * a[i + offset].val();
auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
auto na2 = mod2 - a2;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
}
if (s + 1 != (1 << len)) rot *= info.rate3[bsf(~(unsigned int)(s))];
}
len += 2;
}
}
}
template <class mint>
void butterfly(std::vector<mint>& a) {
butterfly(a.begin(), a.end());
}
template <class It, internal::is_static_modint_t<val_t<It>>* = nullptr>
void butterfly_inv(It a, It last) {
using mint = val_t<It>;
int n = last - a;
int h = internal::ceil_pow2(n);
static constexpr fft_info<mint> info;
int len = h; // a[i, i+(n>>len), i+2*(n>>len), ..] is transformed
while (len) {
if (len == 1) {
int p = 1 << (h - len);
mint irot = 1;
for (int s = 0; s < (1 << (len - 1)); s++) {
int offset = s << (h - len + 1);
for (int i = 0; i < p; i++) {
auto l = a[i + offset];
auto r = a[i + offset + p];
a[i + offset] = l + r;
a[i + offset + p] =
(unsigned long long)(mint::mod() + l.val() - r.val()) *
irot.val();
}
if (s + 1 != (1 << (len - 1)))
irot *= info.irate2[bsf(~(unsigned int)(s))];
}
len--;
} else {
// 4-base
int p = 1 << (h - len);
mint irot = 1, iimag = info.iroot[2];
for (int s = 0; s < (1 << (len - 2)); s++) {
mint irot2 = irot * irot;
mint irot3 = irot2 * irot;
int offset = s << (h - len + 2);
for (int i = 0; i < p; i++) {
auto a0 = 1ULL * a[i + offset + 0 * p].val();
auto a1 = 1ULL * a[i + offset + 1 * p].val();
auto a2 = 1ULL * a[i + offset + 2 * p].val();
auto a3 = 1ULL * a[i + offset + 3 * p].val();
auto a2na3iimag =
1ULL * mint((mint::mod() + a2 - a3) * iimag.val()).val();
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + 1 * p] =
(a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
a[i + offset + 2 * p] =
(a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) * irot2.val();
a[i + offset + 3 * p] =
(a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) *
irot3.val();
}
if (s + 1 != (1 << (len - 2)))
irot *= info.irate3[bsf(~(unsigned int)(s))];
}
len -= 2;
}
}
}
template <class mint>
void butterfly_inv(vector<mint>& a) {
butterfly_inv(a.begin(), a.end());
}
template <class T>
std::vector<T> convolution_naive(
const std::vector<T>& a, const std::vector<T>& b) {
int n = int(a.size()), m = int(b.size());
std::vector<T> ans(n + m - 1);
if (n < m) {
for (int j = 0; j < m; j++) {
for (int i = 0; i < n; i++) {
ans[i + j] += a[i] * b[j];
}
}
} else {
for (int i = 0; i < n; i++) {
for (int j = 0; j < m; j++) {
ans[i + j] += a[i] * b[j];
}
}
}
return ans;
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_fft(std::vector<mint> a, std::vector<mint> b) {
int n = int(a.size()), m = int(b.size());
int z = 1 << internal::ceil_pow2(n + m - 1);
assert((mint::mod() - 1) % z == 0);
a.resize(z);
internal::butterfly(a);
b.resize(z);
internal::butterfly(b);
for (int i = 0; i < z; i++) {
a[i] *= b[i];
}
internal::butterfly_inv(a);
a.resize(n + m - 1);
mint iz = mint(z).inv();
for (int i = 0; i < n + m - 1; i++) a[i] *= iz;
return a;
}
} // namespace internal
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(std::vector<mint>&& a, std::vector<mint>&& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
if (std::min(n, m) <= 60) return convolution_naive(a, b);
return internal::convolution_fft(a, b);
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution(
const std::vector<mint>& a, const std::vector<mint>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
if (std::min(n, m) <= 60) return convolution_naive(a, b);
return internal::convolution_fft(a, b);
}
template <
unsigned int mod = 998244353, class T,
std::enable_if_t<internal::is_integral<T>::value>* = nullptr>
std::vector<T> convolution(const std::vector<T>& a, const std::vector<T>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
using mint = static_modint<mod>;
std::vector<mint> a2(n), b2(m);
for (int i = 0; i < n; i++) {
a2[i] = mint(a[i]);
}
for (int i = 0; i < m; i++) {
b2[i] = mint(b[i]);
}
auto c2 = convolution(move(a2), move(b2));
std::vector<T> c(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
c[i] = c2[i].val();
}
return c;
}
template <class mint>
std::vector<mint> convolution_garner(
const std::vector<mint>& a, const std::vector<mint>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
if (std::min(n, m) <= 200) return internal::convolution_naive(a, b);
static constexpr ll MOD = mint::mod();
static constexpr ll MOD1 = 469762049;
static constexpr ll MOD2 = 167772161;
static constexpr ll MOD3 = 754974721;
static constexpr ll MOD12 = MOD1 * MOD2 % MOD;
static constexpr ll r12 = internal::inv_gcd(MOD1, MOD2).second;
static constexpr ll r23 = internal::inv_gcd(MOD2, MOD3).second;
static constexpr ll r123 = internal::inv_gcd(MOD1 * MOD2, MOD3).second;
vector<int> ai(n), bi(m);
rep(i, n) ai[i] = a[i].val();
rep(i, m) bi[i] = b[i].val();
auto c1 = convolution<MOD1>(ai, bi);
auto c2 = convolution<MOD2>(ai, bi);
auto c3 = convolution<MOD3>(ai, bi);
std::vector<mint> c(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
long long x1 = c1[i];
long long x2 = internal::safe_mod((c2[i] - x1) * r12, MOD2);
long long x3 = internal::safe_mod((c3[i] - x1) * r123 - x2 * r23, MOD3);
c[i] = x1 + x2 * MOD1 + x3 * MOD12;
}
return c;
}
std::vector<long long> convolution_ll(
const std::vector<long long>& a, const std::vector<long long>& b) {
int n = int(a.size()), m = int(b.size());
if (!n || !m) return {};
if (std::min(n, m) <= 200) return internal::convolution_naive(a, b);
static constexpr unsigned long long MOD1 = 754974721; // 2^24
static constexpr unsigned long long MOD2 = 167772161; // 2^25
static constexpr unsigned long long MOD3 = 469762049; // 2^26
static constexpr unsigned long long M2M3 = MOD2 * MOD3;
static constexpr unsigned long long M1M3 = MOD1 * MOD3;
static constexpr unsigned long long M1M2 = MOD1 * MOD2;
static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3;
static constexpr unsigned long long i1 =
internal::inv_gcd(MOD2 * MOD3, MOD1).second;
static constexpr unsigned long long i2 =
internal::inv_gcd(MOD1 * MOD3, MOD2).second;
static constexpr unsigned long long i3 =
internal::inv_gcd(MOD1 * MOD2, MOD3).second;
auto c1 = convolution<MOD1>(a, b);
auto c2 = convolution<MOD2>(a, b);
auto c3 = convolution<MOD3>(a, b);
std::vector<long long> c(n + m - 1);
for (int i = 0; i < n + m - 1; i++) {
unsigned long long x = 0;
x += (c1[i] * i1) % MOD1 * M2M3;
x += (c2[i] * i2) % MOD2 * M1M3;
x += (c3[i] * i3) % MOD3 * M1M2;
// B = 2^63, -B <= x, r(real value) < B
// (x, x - M, x - 2M, or x - 3M) = r (mod 2B)
// r = c1[i] (mod MOD1)
// focus on MOD1
// r = x, x - M', x - 2M', x - 3M' (M' = M % 2^64) (mod 2B)
// r = x,
// x - M' + (0 or 2B),
// x - 2M' + (0, 2B or 4B),
// x - 3M' + (0, 2B, 4B or 6B) (without mod!)
// (r - x) = 0, (0)
// - M' + (0 or 2B), (1)
// -2M' + (0 or 2B or 4B), (2)
// -3M' + (0 or 2B or 4B or 6B) (3) (mod MOD1)
// we checked that
// ((1) mod MOD1) mod 5 = 2
// ((2) mod MOD1) mod 5 = 3
// ((3) mod MOD1) mod 5 = 4
long long diff =
c1[i] - internal::safe_mod((long long)(x), (long long)(MOD1));
if (diff < 0) diff += MOD1;
static constexpr unsigned long long offset[5] = {
0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3};
x -= offset[diff % 5];
c[i] = x;
}
return c;
}
} // namespace atcoder
#endif // ATCODER_CONVOLUTION_HPP
template <class It>
void fft(It a, It last) {
atcoder::internal::butterfly(a, last);
}
template <class T>
void fft(vector<T>& a, int n = -1) {
if (n != -1) a.resize(n);
fft(all(a));
}
template <class It>
void ifft(It a, It last) {
atcoder::internal::butterfly_inv(a, last);
}
template <class T>
void ifft(vector<T>& a) {
ifft(all(a));
}
template <class T>
void double_fft(vector<T>& a) {
static constexpr atcoder::internal::fft_info<T> info{};
int m = a.size();
a.resize(m * 2), copy(a.begin(), a.begin() + m, a.begin() + m);
ifft(a.begin() + m, a.end());
T z = T(m).inv();
T w = info.root[ctz(m * 2)];
rep2(i, m, m * 2) a[i] *= z, z *= w;
fft(a.begin() + m, a.end());
}
#line 3 "ps/inv.hpp"
template <class T>
vector<T> inv(const vector<T>& f, int deg = -1) {
assert(f[0] != 0);
if (deg == -1) deg = f.size();
int z = 1 << atcoder::internal::ceil_pow2(deg);
vector<T> g = {1 / f[0]}, gfft(1);
vector<T> h;
g.reserve(z), gfft.reserve(z), h.reserve(z);
const T i4 = T(4).inv();
T imm4 = i4;
for (int m = 1; m < z; m <<= 1, imm4 *= i4) {
h.assign(f.begin(), f.begin() + min((int)f.size(), m * 2));
copy(all(g), gfft.begin());
fft(h, m * 2), fft(gfft, m * 2);
rep(i, m * 2) h[i] *= gfft[i];
ifft(h), fill(h.begin(), h.begin() + m, 0), fft(h);
rep(i, m * 2) h[i] *= -gfft[i];
ifft(h);
rep2(i, m, m * 2) g.push_back(h[i] * imm4);
}
g.resize(deg);
return g;
}
#line 8 "ps/fps.hpp"
template <class T>
vector<T> exp(const vector<T>& p, int deg = -1);
template <class T>
T bostan_mori(vector<T> a, vector<T> b, ll n);
template <
class T,
enable_if_t<is_base_of_v<atcoder::internal::modint_base, T>>* = nullptr>
class formal_power_series : public vector<T> {
private:
using fps = formal_power_series<T>;
public:
using sparse = vector<pair<int, T>>;
using vector<T>::vector;
formal_power_series(vector<T> v) : vector<T>(move(v)) { }
formal_power_series(sparse p) : vector<T>() {
this->resize(p.back().first + 1);
for (auto [k, c] : p) (*this)[k] += c;
}
static fps one() { return fps({T(1)}); }
static fps zero() { return fps{}; }
int size() const { return vector<T>::size(); }
const vector<T>& as_vec() const { return (const vector<T>&)*this; }
vector<T>& as_vec() { return (vector<T>&)*this; }
vector<T> into_vec() { return move(as_vec()); }
T eval(T x) const {
T pow(1), ans(0);
for (auto e : *this) ans += e * pow, pow *= x;
return ans;
}
void trunc() {
while (!this->empty() && this->back() == T(0)) this->pop_back();
}
fps& operator*=(T c) {
for (auto& e : *this) e *= c;
return *this;
}
fps operator*(T c) && { return move(*this *= c); }
fps operator*(T c) const& { return fps(*this) * c; }
fps mul(T c) && { return move(*this) * c; }
fps mul(T c) const& { return *this * c; }
fps& operator+=(const fps& v) {
this->resize(max(size(), v.size()));
rep(i, v.size()) (*this)[i] += v[i];
return *this;
}
fps operator+(const fps& v) && { return move(*this += v); }
fps operator+(const fps& v) const& { return fps(*this) + v; }
fps add(const fps& v) && { return move(*this += v); }
fps add(const fps& v) const& { return fps(*this) + v; }
fps& operator-=(const fps& v) {
this->resize(max(size(), v.size()));
rep(i, v.size()) (*this)[i] -= v[i];
return *this;
}
fps operator-(const fps& v) && { return move(*this -= v); }
fps operator-(const fps& v) const& { return fps(*this) - v; }
fps sub(const fps& v) && { return move(*this -= v); }
fps sub(const fps& v) const& { return fps(*this) - v; }
fps conv(fps v, int deg = -1) && {
if (~deg) this->resize(min(size(), deg)), v.resize(min(v.size(), deg));
auto f = convolution(into_vec(), v.into_vec());
if (~deg) f.resize(deg);
return f;
}
fps conv(fps v, int deg = -1) const& { return fps(*this).conv(move(v), deg); }
fps diff() && {
rep(i, size() - 1) (*this)[i] = (*this)[i + 1] * (i + 1);
this->pop_back();
return move(*this);
}
fps diff() const& { return fps(*this).diff(); }
fps integr() && {
this->push_back(0);
repr(i, size() - 1) (*this)[i + 1] = (*this)[i] * inverse(i + 1);
(*this)[0] = 0;
return move(*this);
}
fps integr() const& { return fps(*this).integr(); }
fps inv(int deg = -1) const& { return fps(::inv(as_vec(), deg)); }
fps div(const fps& v, int deg = -1) && {
return move(*this).conv(v.inv(deg), deg);
}
fps div(const fps& v, int deg = -1) const& { return fps(*this).div(v, deg); }
fps log(int deg = -1) && {
return inv(deg - 1).conv(move(*this).diff(), deg - 1).integr();
}
fps log(int deg = -1) const& { return fps(*this).log(deg); }
fps exp(int deg = -1) const& { return ::exp(as_vec(), deg); }
fps pow(ll k, int deg = -1) && {
if (deg == -1) deg = size();
int z = -1;
rep(i, size())
if ((*this)[i] != 0) {
z = i;
break;
}
if (z == -1 || sat_mul<ll>(z, k) > deg) {
fps res(deg, 0);
res[0] = k == 0;
return res;
}
ll rest = deg - z * k;
this->erase(this->begin(), this->begin() + z);
T c = (*this)[0].pow(k);
fps f = move(*this).log(rest).mul(k).exp(rest);
for (auto& e : f) e *= c;
f.resize(deg);
copy_backward(f.begin(), f.begin() + rest, f.end());
fill(f.begin(), f.begin() + z * k, 0);
return f;
}
fps pow(ll k, int deg = -1) const& { return fps(*this).pow(k, deg); }
fps square(int deg = -1) && {
if (deg == -1) deg = size() * 2 - 1;
int n = 1 << atcoder::internal::ceil_pow2(deg);
fft(as_vec(), n * 2);
for (auto& e : *this) e *= e;
ifft(as_vec());
auto in2 = inverse(n * 2);
this->resize(deg);
for (auto& e : *this) e *= in2;
return move(*this);
}
fps square(int deg = -1) const& { return fps(*this).square(deg); }
T div_at(fps f, ll n) && { return bostan_mori(into_vec(), f.into_vec(), n); }
T div_at(fps f, ll n) const& { return fps(*this).div_at(move(f), n); }
optional<fps> sqrt(int deg = -1) && {
if (deg == -1) deg = size();
this->resize(deg);
if (this->empty()) return move(*this);
if ((*this)[0] == 0) {
int b = 0;
while (b < size() && (*this)[b] == 0) b++;
if (b == size()) return move(*this);
if (b % 2 != 0) return nullopt;
this->erase(this->begin(), this->begin() + b);
auto ans = move(*this).sqrt(deg - b / 2);
if (ans) ans->insert(ans->begin(), b / 2, T(0));
return ans;
}
auto x = mod_sqrt((*this)[0]);
if (!x) return nullopt;
fps f = {*x};
int z = 1 << atcoder::internal::ceil_pow2(deg);
f.reserve(z);
const T i2 = inverse(2);
for (int m = 1; m < z; m *= 2) {
fps h(this->begin(), this->begin() + min(m * 2, size()));
fps hf = move(h).div(f, m * 2);
f = move(f).add(hf).mul(i2);
}
f.resize(deg);
return f;
}
fps div_poly(fps g) && {
int d = size() - g.size() + 1;
if (d <= 0) return zero();
reverse(all(*this));
reverse(all(g));
fps q = move(*this).div(move(g), d);
reverse(all(q));
return q;
}
fps div_poly(fps g) const& { return fps(*this).div_poly(move(g)); }
pair<fps, fps> div_rem_poly(fps g) && {
int d = g.size() - 1;
fps q = div_poly(g);
fps r = move(*this).sub(move(g).conv(q, d));
r.resize(d);
r.trunc();
return pair(move(q), move(r));
}
pair<fps, fps> div_rem_poly(fps g) const& {
return fps(*this).div_rem_poly(move(g));
}
fps conv(sparse v) && {
if (v.empty()) return zero();
if (v.front().first == 0) v.front().second -= T(1);
repr(i, size())
for (auto [k, c] : v) {
if (k > i) break;
(*this)[i] += (*this)[i - k] * c;
}
return move(*this);
}
fps conv(sparse v) const& { return fps(*this).conv(move(v)); }
fps div(sparse v) && {
auto [k0, r] = v.front();
assert(k0 == 0 && r != T(0));
T ir = r.inv();
v.erase(v.begin());
rep(i, size()) {
for (auto [k, c] : v) {
if (k > i) break;
(*this)[i] -= (*this)[i - k] * c;
}
(*this)[i] *= ir;
}
return move(*this);
}
fps div(sparse v) const& { return fps(*this).div(move(v)); }
template <class It>
static fps prod(It first, It last) {
if (first == last) return one();
vector<fps> vec(first, last);
vec.reserve(distance(first, last) * 2);
for (int i = 0; i + 1 < vec.size(); i += 2)
vec.push_back(move(vec[i]).conv(move(vec[i + 1])));
return vec.back();
}
};
#line 3 "ps/frac.hpp"
template <class T>
class fraction {
public:
using fps = formal_power_series<T>;
fps numer, denom;
fraction() : numer(fps::zero()), denom(fps::one()) { }
fraction(fps numer) : numer(numer), denom(fps::one()) { }
fraction(fps numer, fps denom) : numer(numer), denom(denom) { }
T eval(T x) const { return numer.eval(x) / denom.eval(x); }
fraction add(fraction rhs) && {
numer = move(numer).conv(rhs.denom).add(move(rhs.numer).conv(denom));
denom = move(denom).conv(move(rhs.denom));
return move(*this);
}
fraction add(fraction rhs) const& { return fraction(*this).add(move(rhs)); }
fraction sub(fraction rhs) && {
numer = move(numer).conv(rhs.denom).sub(move(rhs.numer).conv(denom));
denom = move(denom).conv(move(rhs.denom));
return move(*this);
}
fraction sub(fraction rhs) const& { return fraction(*this).sub(move(rhs)); }
fraction mul(fraction rhs) && {
numer = move(numer).conv(move(rhs.numer));
denom = move(denom).conv(move(rhs.denom));
return move(*this);
}
fraction mul(fraction rhs) const& { return fraction(*this).mul(move(rhs)); }
fraction div(fraction rhs) && {
numer = move(numer).conv(move(rhs.denom));
denom = move(denom).conv(move(rhs.numer));
return move(*this);
}
fraction div(fraction rhs) const& { return fraction(*this).div(move(rhs)); }
fraction diff() && {
fps pd = numer.diff(), qd = denom.diff();
numer = move(pd).conv(denom).sub(move(numer).conv(move(qd)));
denom = move(denom).square();
return move(*this);
}
fraction diff() const& { return fraction(*this).diff(); }
};