formal_power_series.hpp¶
Formal power series logarithm, exponential, power, and square root operations.
Verified by exp_of_formal_power_series, inv_of_formal_power_series, log_of_formal_power_series, pow_of_formal_power_series, sqrt_of_formal_power_series.
\[
\displaystyle \log f=\int\frac{f'}{f},\qquad \exp g=f\iff \log f=g
\]
Implementation¶
#ifndef NOYA_FORMAL_POWER_SERIES_HPP
#define NOYA_FORMAL_POWER_SERIES_HPP 1
/// @complexity Time: O(M(n) log n) for inverse/log/exp with convolution cost M(n).
/// Space: O(n log n) temporary coefficients.
#include "noya/mod_sqrt.hpp"
#include "noya/polynomial.hpp"
#include <algorithm>
#include <cassert>
#include <cstdint>
#include <optional>
#include <vector>
namespace noya {
/// @brief Return the first n coefficients of 1/f; requires f[0] != 0.
template <class Mint>
std::vector<Mint> fps_inverse(const std::vector<Mint> &f, int n) {
return polynomial_inverse_series(f, n);
}
/// @brief Return log(f) modulo x^n; requires f[0] = 1.
template <class Mint>
std::vector<Mint> fps_logarithm(const std::vector<Mint> &f, int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
assert(!f.empty() && f[0] == Mint(1));
std::vector<Mint> derivative = polynomial_derivative(f);
std::vector<Mint> inverse = fps_inverse(f, n);
std::vector<Mint> product = atcoder::convolution(derivative, inverse);
product.resize(n - 1);
std::vector<Mint> result = polynomial_integral(product);
result.resize(n);
return result;
}
/// @brief Return exp(f) modulo x^n; requires f[0] = 0.
template <class Mint>
std::vector<Mint> fps_exponential(const std::vector<Mint> &f, int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
assert(f.empty() || f[0] == Mint{});
std::vector<Mint> result = {Mint(1)};
while (int(result.size()) < n) {
int target = std::min(n, int(result.size()) * 2);
std::vector<Mint> logarithm = fps_logarithm(result, target);
std::vector<Mint> correction(target);
for (int i = 0; i < target; i++) {
if (i < int(f.size())) {
correction[i] += f[i];
}
correction[i] -= logarithm[i];
}
correction[0] += Mint(1);
result = atcoder::convolution(result, correction);
result.resize(target);
}
return result;
}
/// @brief Compute value^exponent by binary exponentiation.
template <class Mint>
Mint fps_scalar_power(Mint value, std::uint64_t exponent) {
Mint result = Mint(1);
while (exponent > 0) {
if (exponent & 1) {
result *= value;
}
value *= value;
exponent >>= 1;
}
return result;
}
/// @brief Return f^exponent modulo x^n for a nonnegative exponent.
template <class Mint>
std::vector<Mint> fps_power(const std::vector<Mint> &f, std::uint64_t exponent,
int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
std::vector<Mint> zero(n);
if (exponent == 0) {
zero[0] = Mint(1);
return zero;
}
int first = 0;
while (first < int(f.size()) && f[first] == Mint{}) {
first++;
}
if (first == int(f.size()) ||
(first > 0 && exponent > std::uint64_t((n - 1) / first))) {
return zero;
}
int shift = int(std::uint64_t(first) * exponent);
int target = n - shift;
Mint leading = f[first];
std::vector<Mint> normalized(target);
for (int i = 0; i < target && first + i < int(f.size()); i++) {
normalized[i] = f[first + i] / leading;
}
std::vector<Mint> logarithm = fps_logarithm(normalized, target);
Mint scalar_exponent = Mint(exponent);
for (Mint &value : logarithm) {
value *= scalar_exponent;
}
std::vector<Mint> powered = fps_exponential(logarithm, target);
Mint leading_power = fps_scalar_power(leading, exponent);
for (Mint &value : powered) {
value *= leading_power;
}
std::vector<Mint> result(n);
for (int i = 0; i < target; i++) {
result[shift + i] = powered[i];
}
return result;
}
/// @brief Return a formal square root of f modulo x^n, if one exists.
template <class Mint>
std::optional<std::vector<Mint>> fps_square_root(const std::vector<Mint> &f,
int n) {
assert(n >= 0);
if (n == 0) {
return std::vector<Mint>{};
}
int first = 0;
while (first < std::min(n, int(f.size())) && f[first] == Mint{}) {
first++;
}
if (first == std::min(n, int(f.size()))) {
return std::vector<Mint>(n);
}
if (first & 1) {
return std::nullopt;
}
if (first > 0) {
int shift = first / 2;
int target = n - first;
std::vector<Mint> reduced(target);
for (int i = 0; i < target && first + i < int(f.size()); i++) {
reduced[i] = f[first + i];
}
auto root = fps_square_root(reduced, target);
if (!root) {
return std::nullopt;
}
std::vector<Mint> result(n);
for (int i = 0; i < int(root->size()) && shift + i < n; i++) {
result[shift + i] = (*root)[i];
}
return result;
}
auto constant_root =
mod_sqrt(std::uint64_t(f[0].val()), std::uint64_t(Mint::mod()));
if (!constant_root) {
return std::nullopt;
}
std::vector<Mint> result = {Mint(*constant_root)};
Mint inverse_two = Mint(1) / Mint(2);
while (int(result.size()) < n) {
int target = std::min(n, int(result.size()) * 2);
std::vector<Mint> prefix(target);
for (int i = 0; i < target && i < int(f.size()); i++) {
prefix[i] = f[i];
}
std::vector<Mint> quotient = atcoder::convolution(
prefix, polynomial_inverse_series(result, target));
quotient.resize(target);
result.resize(target);
for (int i = 0; i < target; i++) {
result[i] = (result[i] + quotient[i]) * inverse_two;
}
}
return result;
}
} // namespace noya
#endif // NOYA_FORMAL_POWER_SERIES_HPP
#include <algorithm>
#include <array>
#include <cassert>
#include <cstdint>
#include <numeric>
#include <optional>
#include <type_traits>
#include <utility>
#include <vector>
/// @complexity Time: O(M(n) log n) for inverse/log/exp with convolution cost M(n).
/// Space: O(n log n) temporary coefficients.
/// @complexity Time: O(log^2 p).
/// Space: O(1).
/// @complexity Time: O(log^3 n) primality testing; Pollard-rho factorization is expected about O(n^(1/4)).
/// Space: O(log n) recursion and factors.
namespace noya {
namespace factorize_internal {
using u64 = std::uint64_t;
using u128 = unsigned __int128;
inline u64 multiply_mod(u64 a, u64 b, u64 mod) {
return u64(u128(a) * b % mod);
}
inline u64 power_mod(u64 a, u64 exponent, u64 mod) {
u64 result = 1;
while (exponent > 0) {
if (exponent & 1) {
result = multiply_mod(result, a, mod);
}
a = multiply_mod(a, a, mod);
exponent >>= 1;
}
return result;
}
inline bool miller_rabin(u64 n) {
if (n < 2) {
return false;
}
for (u64 p :
std::array<u64, 12>{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37}) {
if (n % p == 0) {
return n == p;
}
}
int shift = __builtin_ctzll(n - 1);
u64 odd = (n - 1) >> shift;
for (u64 base :
std::array<u64, 7>{2, 325, 9375, 28178, 450775, 9780504, 1795265022}) {
if (base % n == 0) {
continue;
}
u64 value = power_mod(base % n, odd, n);
if (value == 1 || value == n - 1) {
continue;
}
bool composite = true;
for (int i = 1; i < shift; i++) {
value = multiply_mod(value, value, n);
if (value == n - 1) {
composite = false;
break;
}
}
if (composite) {
return false;
}
}
return true;
}
inline u64 splitmix64(u64 &state) {
u64 z = (state += 0x9e3779b97f4a7c15ULL);
z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9ULL;
z = (z ^ (z >> 27)) * 0x94d049bb133111ebULL;
return z ^ (z >> 31);
}
inline u64 pollard_rho(u64 n) {
if (n % 2 == 0) {
return 2;
}
if (n % 3 == 0) {
return 3;
}
static u64 state = 0x123456789abcdef0ULL;
while (true) {
u64 y = splitmix64(state) % (n - 1) + 1;
u64 c = splitmix64(state) % (n - 1) + 1;
constexpr u64 block = 128;
u64 g = 1;
u64 r = 1;
u64 q = 1;
u64 x = 0;
u64 saved_y = 0;
auto next = [&](u64 value) {
return u64((u128(multiply_mod(value, value, n)) + c) % n);
};
while (g == 1) {
x = y;
for (u64 i = 0; i < r; i++) {
y = next(y);
}
for (u64 offset = 0; offset < r && g == 1; offset += block) {
saved_y = y;
for (u64 i = 0; i < std::min(block, r - offset); i++) {
y = next(y);
u64 difference = x > y ? x - y : y - x;
q = multiply_mod(q, difference, n);
}
g = std::gcd(q, n);
}
r <<= 1;
}
if (g == n) {
do {
saved_y = next(saved_y);
u64 difference = x > saved_y ? x - saved_y : saved_y - x;
g = std::gcd(difference, n);
} while (g == 1);
}
if (g != n) {
return g;
}
}
}
inline void collect_factors(u64 n, std::vector<u64> &result) {
if (n == 1) {
return;
}
if (miller_rabin(n)) {
result.push_back(n);
return;
}
u64 factor = pollard_rho(n);
collect_factors(factor, result);
collect_factors(n / factor, result);
}
} // namespace factorize_internal
/// @brief Deterministic Miller-Rabin primality test for unsigned 64-bit
/// integers.
inline bool is_prime(std::uint64_t n) {
return factorize_internal::miller_rabin(n);
}
/// @brief Return the prime factors of n with multiplicity in increasing order.
inline std::vector<std::uint64_t> prime_factors(std::uint64_t n) {
assert(n >= 1);
std::vector<std::uint64_t> result;
factorize_internal::collect_factors(n, result);
std::sort(result.begin(), result.end());
return result;
}
/// @brief Return the prime factorization of n as (prime, exponent) pairs.
inline std::vector<std::pair<std::uint64_t, int>> factorize(std::uint64_t n) {
std::vector<std::pair<std::uint64_t, int>> result;
for (std::uint64_t p : prime_factors(n)) {
if (result.empty() || result.back().first != p) {
result.emplace_back(p, 1);
} else {
result.back().second++;
}
}
return result;
}
} // namespace noya
namespace noya {
/// @brief Compute the smaller square root modulo a prime, or nullopt if no
/// square root exists.
inline std::optional<std::uint64_t> mod_sqrt(std::uint64_t value,
std::uint64_t modulus) {
assert(modulus >= 2 && is_prime(modulus));
value %= modulus;
if (modulus == 2 || value == 0) {
return value;
}
using factorize_internal::multiply_mod;
using factorize_internal::power_mod;
if (power_mod(value, (modulus - 1) / 2, modulus) != 1) {
return std::nullopt;
}
if (modulus % 4 == 3) {
std::uint64_t root = power_mod(value, (modulus + 1) / 4, modulus);
return std::min(root, modulus - root);
}
std::uint64_t odd = modulus - 1;
int exponent = 0;
while ((odd & 1) == 0) {
odd >>= 1;
exponent++;
}
std::uint64_t non_residue = 2;
while (power_mod(non_residue, (modulus - 1) / 2, modulus) != modulus - 1) {
non_residue++;
}
std::uint64_t root = power_mod(value, (odd + 1) / 2, modulus);
std::uint64_t remainder = power_mod(value, odd, modulus);
std::uint64_t step = power_mod(non_residue, odd, modulus);
int remaining = exponent;
while (remainder != 1) {
std::uint64_t squared = remainder;
int shift = 0;
while (squared != 1 && shift < remaining) {
squared = multiply_mod(squared, squared, modulus);
shift++;
}
assert(shift < remaining);
std::uint64_t multiplier =
power_mod(step, std::uint64_t(1) << (remaining - shift - 1), modulus);
root = multiply_mod(root, multiplier, modulus);
step = multiply_mod(multiplier, multiplier, modulus);
remainder = multiply_mod(remainder, step, modulus);
remaining = shift;
}
return std::min(root, modulus - root);
}
} // namespace noya
/// @complexity Time: O(M(n) log n) inverse/division and O(M(n)) Taylor shift.
/// Space: O(n log n) temporaries.
#ifdef _MSC_VER
#include <intrin.h>
#endif
#if __cplusplus >= 202002L
#include <bit>
#endif
namespace atcoder {
namespace internal {
#if __cplusplus >= 202002L
using std::bit_ceil;
#else
// @return same with std::bit::bit_ceil
unsigned int bit_ceil(unsigned int n) {
unsigned int x = 1;
while (x < (unsigned int)(n)) x *= 2;
return x;
}
#endif
// @param n `1 <= n`
// @return same with std::bit::countr_zero
int countr_zero(unsigned int n) {
#ifdef _MSC_VER
unsigned long index;
_BitScanForward(&index, n);
return index;
#else
return __builtin_ctz(n);
#endif
}
// @param n `1 <= n`
// @return same with std::bit::countr_zero
constexpr int countr_zero_constexpr(unsigned int n) {
int x = 0;
while (!(n & (1 << x))) x++;
return x;
}
} // namespace internal
} // namespace atcoder
#ifdef _MSC_VER
#include <intrin.h>
#endif
#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`
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 long long y = x * _m;
return (unsigned int)(z - y + (z < y ? _m : 0));
}
};
// @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
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
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 mint raw(int v) {
mint x;
x._v = v;
return x;
}
static_modint() : _v(0) {}
template <class T, internal::is_signed_int_t<T>* = nullptr>
static_modint(T 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>
static_modint(T v) {
_v = (unsigned int)(v % umod());
}
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 -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint& operator*=(const mint& rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
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 {
if (prime) {
assert(_v);
return pow(umod() - 2);
} else {
auto eg = internal::inv_gcd(_v, m);
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;
}
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());
}
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;
}
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
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 = countr_zero_constexpr(mint::mod() - 1);
std::array<mint, rank2 + 1> root; // root[i]^(2^i) == 1
std::array<mint, rank2 + 1> iroot; // root[i] * iroot[i] == 1
std::array<mint, std::max(0, rank2 - 2 + 1)> rate2;
std::array<mint, std::max(0, rank2 - 2 + 1)> irate2;
std::array<mint, std::max(0, rank2 - 3 + 1)> rate3;
std::array<mint, std::max(0, rank2 - 3 + 1)> irate3;
fft_info() {
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 mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly(std::vector<mint>& a) {
int n = int(a.size());
int h = internal::countr_zero((unsigned int)n);
static const 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[countr_zero(~(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[countr_zero(~(unsigned int)(s))];
}
len += 2;
}
}
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
void butterfly_inv(std::vector<mint>& a) {
int n = int(a.size());
int h = internal::countr_zero((unsigned int)n);
static const 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)((unsigned int)(l.val() - r.val()) + mint::mod()) *
irot.val();
;
}
if (s + 1 != (1 << (len - 1)))
irot *= info.irate2[countr_zero(~(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[countr_zero(~(unsigned int)(s))];
}
len -= 2;
}
}
}
template <class mint, internal::is_static_modint_t<mint>* = nullptr>
std::vector<mint> convolution_naive(const std::vector<mint>& a,
const std::vector<mint>& b) {
int n = int(a.size()), m = int(b.size());
std::vector<mint> 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 = (int)internal::bit_ceil((unsigned int)(n + m - 1));
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 {};
int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
assert((mint::mod() - 1) % z == 0);
if (std::min(n, m) <= 60) return convolution_naive(std::move(a), std::move(b));
return internal::convolution_fft(std::move(a), std::move(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 {};
int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
assert((mint::mod() - 1) % z == 0);
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>;
int z = (int)internal::bit_ceil((unsigned int)(n + m - 1));
assert((mint::mod() - 1) % z == 0);
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(std::move(a2), std::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;
}
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 {};
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;
static constexpr int MAX_AB_BIT = 24;
static_assert(MOD1 % (1ull << MAX_AB_BIT) == 1, "MOD1 isn't enough to support an array length of 2^24.");
static_assert(MOD2 % (1ull << MAX_AB_BIT) == 1, "MOD2 isn't enough to support an array length of 2^24.");
static_assert(MOD3 % (1ull << MAX_AB_BIT) == 1, "MOD3 isn't enough to support an array length of 2^24.");
assert(n + m - 1 <= (1 << MAX_AB_BIT));
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
namespace noya {
/// @brief Remove trailing zero coefficients from a polynomial.
template <class T> void polynomial_trim(std::vector<T> &polynomial) {
while (!polynomial.empty() && polynomial.back() == T{}) {
polynomial.pop_back();
}
}
/// @brief Return the formal derivative of a polynomial.
template <class T>
std::vector<T> polynomial_derivative(const std::vector<T> &polynomial) {
if (polynomial.size() <= 1) {
return {};
}
std::vector<T> result(polynomial.size() - 1);
for (int i = 1; i < int(polynomial.size()); i++) {
result[i - 1] = polynomial[i] * T(i);
}
return result;
}
/// @brief Return the formal integral with constant coefficient zero.
template <class T>
std::vector<T> polynomial_integral(const std::vector<T> &polynomial) {
std::vector<T> result(polynomial.size() + 1);
for (int i = 0; i < int(polynomial.size()); i++) {
result[i + 1] = polynomial[i] / T(i + 1);
}
return result;
}
/// @brief Return the first n coefficients of 1/f using Newton iteration.
template <class Mint>
std::vector<Mint> polynomial_inverse_series(const std::vector<Mint> &f, int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
assert(!f.empty() && f[0] != Mint{});
std::vector<Mint> inverse = {Mint(1) / f[0]};
while (int(inverse.size()) < n) {
int target = std::min(n, int(inverse.size()) * 2);
std::vector<Mint> prefix(target);
for (int i = 0; i < std::min(target, int(f.size())); i++) {
prefix[i] = f[i];
}
std::vector<Mint> correction = atcoder::convolution(prefix, inverse);
correction.resize(target);
for (Mint &value : correction) {
value = -value;
}
correction[0] += Mint(2);
inverse = atcoder::convolution(inverse, correction);
inverse.resize(target);
}
return inverse;
}
/// @brief Divide f by nonzero g and return (quotient, remainder).
template <class Mint>
std::pair<std::vector<Mint>, std::vector<Mint>>
polynomial_divmod(std::vector<Mint> f, std::vector<Mint> g) {
polynomial_trim(f);
polynomial_trim(g);
assert(!g.empty());
if (f.size() < g.size()) {
return {{}, f};
}
int quotient_size = int(f.size() - g.size() + 1);
std::vector<Mint> reversed_f(f.rbegin(), f.rend());
std::vector<Mint> reversed_g(g.rbegin(), g.rend());
reversed_f.resize(quotient_size);
reversed_g.resize(quotient_size);
std::vector<Mint> inverse =
polynomial_inverse_series(reversed_g, quotient_size);
std::vector<Mint> quotient = atcoder::convolution(reversed_f, inverse);
quotient.resize(quotient_size);
std::reverse(quotient.begin(), quotient.end());
std::vector<Mint> product = atcoder::convolution(quotient, g);
for (int i = 0; i < int(product.size()); i++) {
f[i] -= product[i];
}
polynomial_trim(f);
return {quotient, f};
}
/// @brief Return f(x + shift) in O(M(n)) time.
template <class Mint>
std::vector<Mint> polynomial_taylor_shift(const std::vector<Mint> &f,
Mint shift) {
int size = int(f.size());
if (size == 0) {
return {};
}
std::vector<Mint> factorial(size, Mint(1));
std::vector<Mint> inverse_factorial(size, Mint(1));
for (int i = 1; i < size; i++) {
factorial[i] = factorial[i - 1] * Mint(i);
}
inverse_factorial.back() = Mint(1) / factorial.back();
for (int i = size - 1; i > 0; i--) {
inverse_factorial[i - 1] = inverse_factorial[i] * Mint(i);
}
std::vector<Mint> reversed(size), powers(size);
Mint power = Mint(1);
for (int i = 0; i < size; i++) {
reversed[size - 1 - i] = f[i] * factorial[i];
powers[i] = power * inverse_factorial[i];
power *= shift;
}
std::vector<Mint> product = atcoder::convolution(reversed, powers);
std::vector<Mint> result(size);
for (int i = 0; i < size; i++) {
result[i] = product[size - 1 - i] * inverse_factorial[i];
}
return result;
}
} // namespace noya
namespace noya {
/// @brief Return the first n coefficients of 1/f; requires f[0] != 0.
template <class Mint>
std::vector<Mint> fps_inverse(const std::vector<Mint> &f, int n) {
return polynomial_inverse_series(f, n);
}
/// @brief Return log(f) modulo x^n; requires f[0] = 1.
template <class Mint>
std::vector<Mint> fps_logarithm(const std::vector<Mint> &f, int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
assert(!f.empty() && f[0] == Mint(1));
std::vector<Mint> derivative = polynomial_derivative(f);
std::vector<Mint> inverse = fps_inverse(f, n);
std::vector<Mint> product = atcoder::convolution(derivative, inverse);
product.resize(n - 1);
std::vector<Mint> result = polynomial_integral(product);
result.resize(n);
return result;
}
/// @brief Return exp(f) modulo x^n; requires f[0] = 0.
template <class Mint>
std::vector<Mint> fps_exponential(const std::vector<Mint> &f, int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
assert(f.empty() || f[0] == Mint{});
std::vector<Mint> result = {Mint(1)};
while (int(result.size()) < n) {
int target = std::min(n, int(result.size()) * 2);
std::vector<Mint> logarithm = fps_logarithm(result, target);
std::vector<Mint> correction(target);
for (int i = 0; i < target; i++) {
if (i < int(f.size())) {
correction[i] += f[i];
}
correction[i] -= logarithm[i];
}
correction[0] += Mint(1);
result = atcoder::convolution(result, correction);
result.resize(target);
}
return result;
}
/// @brief Compute value^exponent by binary exponentiation.
template <class Mint>
Mint fps_scalar_power(Mint value, std::uint64_t exponent) {
Mint result = Mint(1);
while (exponent > 0) {
if (exponent & 1) {
result *= value;
}
value *= value;
exponent >>= 1;
}
return result;
}
/// @brief Return f^exponent modulo x^n for a nonnegative exponent.
template <class Mint>
std::vector<Mint> fps_power(const std::vector<Mint> &f, std::uint64_t exponent,
int n) {
assert(n >= 0);
if (n == 0) {
return {};
}
std::vector<Mint> zero(n);
if (exponent == 0) {
zero[0] = Mint(1);
return zero;
}
int first = 0;
while (first < int(f.size()) && f[first] == Mint{}) {
first++;
}
if (first == int(f.size()) ||
(first > 0 && exponent > std::uint64_t((n - 1) / first))) {
return zero;
}
int shift = int(std::uint64_t(first) * exponent);
int target = n - shift;
Mint leading = f[first];
std::vector<Mint> normalized(target);
for (int i = 0; i < target && first + i < int(f.size()); i++) {
normalized[i] = f[first + i] / leading;
}
std::vector<Mint> logarithm = fps_logarithm(normalized, target);
Mint scalar_exponent = Mint(exponent);
for (Mint &value : logarithm) {
value *= scalar_exponent;
}
std::vector<Mint> powered = fps_exponential(logarithm, target);
Mint leading_power = fps_scalar_power(leading, exponent);
for (Mint &value : powered) {
value *= leading_power;
}
std::vector<Mint> result(n);
for (int i = 0; i < target; i++) {
result[shift + i] = powered[i];
}
return result;
}
/// @brief Return a formal square root of f modulo x^n, if one exists.
template <class Mint>
std::optional<std::vector<Mint>> fps_square_root(const std::vector<Mint> &f,
int n) {
assert(n >= 0);
if (n == 0) {
return std::vector<Mint>{};
}
int first = 0;
while (first < std::min(n, int(f.size())) && f[first] == Mint{}) {
first++;
}
if (first == std::min(n, int(f.size()))) {
return std::vector<Mint>(n);
}
if (first & 1) {
return std::nullopt;
}
if (first > 0) {
int shift = first / 2;
int target = n - first;
std::vector<Mint> reduced(target);
for (int i = 0; i < target && first + i < int(f.size()); i++) {
reduced[i] = f[first + i];
}
auto root = fps_square_root(reduced, target);
if (!root) {
return std::nullopt;
}
std::vector<Mint> result(n);
for (int i = 0; i < int(root->size()) && shift + i < n; i++) {
result[shift + i] = (*root)[i];
}
return result;
}
auto constant_root =
mod_sqrt(std::uint64_t(f[0].val()), std::uint64_t(Mint::mod()));
if (!constant_root) {
return std::nullopt;
}
std::vector<Mint> result = {Mint(*constant_root)};
Mint inverse_two = Mint(1) / Mint(2);
while (int(result.size()) < n) {
int target = std::min(n, int(result.size()) * 2);
std::vector<Mint> prefix(target);
for (int i = 0; i < target && i < int(f.size()); i++) {
prefix[i] = f[i];
}
std::vector<Mint> quotient = atcoder::convolution(
prefix, polynomial_inverse_series(result, target));
quotient.resize(target);
result.resize(target);
for (int i = 0; i < target; i++) {
result[i] = (result[i] + quotient[i]) * inverse_two;
}
}
return result;
}
} // namespace noya