factorials (階乗, 階乗の逆元, nCr, nPr)
(Math/Combinatorics/factorials.hpp)

#pragma once
#include <cassert>
#include <vector>

#include "../Modulo/mod-int.hpp"

/**
 * @brief factorials (階乗, 階乗の逆元, nCr, nPr)
 */
template <class Mint>
struct Factorials {
public:
    using value_type = Mint;
    static constexpr size_t MAX_N = std::min<size_t>(1e7, Mint::mod()) + 1;

private:
    mutable std::vector<value_type> m_fact, m_finv;

public:
    Factorials() {
        m_fact.reserve(MAX_N);
        m_finv.reserve(MAX_N);
        m_fact.resize(2, value_type::raw(1));   // m_fact[0] = m_fact[1] = 1
        m_finv.resize(2, value_type::raw(1));   // m_finv[0] = m_finv[1] = 1
    }

    void preCalc(size_t n) const {
        if (n < m_fact.size()) return;
        const size_t l = m_fact.size();
        const size_t r = n + 1;
        m_fact.resize(r), m_finv.resize(r);
        for (size_t i = l; i < r; ++i) m_fact[i] = m_fact[i - 1] * i;
        m_finv[r - 1] = m_fact[r - 1].inv();
        for (size_t i = r - 1; i > l; --i) m_finv[i - 1] = m_finv[i] * i;
    }

    const value_type fact(int i) const { return preCalc(i), m_fact[i]; }
    const value_type finv(int i) const { return preCalc(i), m_finv[i]; }

    const value_type C(int n, int r) const {
        if (r < 0 || n < r) return value_type::raw(0);
        return preCalc(n), m_fact[n] * m_finv[r] * m_finv[n - r];
    }

    const value_type P(int n, int r) const {
        if (r < 0 || n < r) return value_type::raw(0);
        return preCalc(n), m_fact[n] * m_finv[n - r];
    }

    const value_type H(int n, int r) const {
        if (n < 0 || r < 0) return value_type::raw(0);
        if (n == 0 && r == 0) return value_type::raw(1);
        return C(n + r - 1, r);
    }
};

#line 2 "Math/Combinatorics/factorials.hpp"
#include <cassert>
#include <vector>

#line 2 "Math/Modulo/mod-int.hpp"
#include <cstdint>
#line 4 "Math/Modulo/mod-int.hpp"
#include <iostream>
#include <limits>

/**
 * @brief Mod-Int (コンパイル時mod型と実行時mod型)
 */
namespace internal {

template <class ModHolder>
class ModInt {
private:
    int64_t value;

public:
    constexpr ModInt() noexcept
        : value(0) {}

    constexpr ModInt(int64_t v) noexcept
        : value(ModInt::normalized(v)) {}

    static constexpr const ModInt raw(int64_t v) noexcept {
        ModInt ret;
        ret.value = v;
        return ret;
    }

    static constexpr ModInt nullopt() noexcept { return ModInt::raw(std::numeric_limits<int64_t>::min()); }

    constexpr bool isNull() const noexcept { return *this == ModInt::nullopt(); }

    static constexpr int64_t mod() { return ModHolder::mod; }

    static int64_t setMod(int64_t m) noexcept {
        assert(m >= 1);
        return ModHolder::mod = m;
    }

    template <class T>
    constexpr explicit operator T() const noexcept {
        return static_cast<T>(value);
    }

    constexpr int64_t val() const noexcept { return value; }

    constexpr ModInt& operator+=(const ModInt& rhs) noexcept {
        if ((value += rhs.value) >= mod()) value -= mod();
        return *this;
    }
    constexpr ModInt& operator-=(const ModInt& rhs) noexcept {
        if ((value -= rhs.value) < 0) value += mod();
        return *this;
    }
    constexpr ModInt& operator*=(const ModInt& rhs) noexcept {
        (value *= rhs.value) %= mod();
        return *this;
    }
    constexpr ModInt& operator/=(const ModInt& rhs) noexcept { return *this *= rhs.inv(); }
    constexpr const ModInt inv() const noexcept { return ModInt(ModInt::inverse(value, mod())); }
    constexpr const ModInt operator+() const noexcept { return *this; }
    constexpr const ModInt operator-() const noexcept { return ModInt(-value); }

    constexpr const ModInt pow(int64_t expv) const noexcept {
        int64_t ret = 1, square = value;
        for (uint64_t p = std::abs(expv); p; p >>= 1) {
            if (p & 1) (ret *= square) %= mod();
            (square *= square) %= mod();
        }
        return (expv < 0) ? (ModInt::raw(1) / ModInt::raw(ret)) : ModInt::raw(ret);
    }

    friend constexpr bool operator==(const ModInt& lhs, const ModInt& rhs) noexcept { return lhs.value == rhs.value; }
    friend constexpr bool operator!=(const ModInt& lhs, const ModInt& rhs) noexcept { return lhs.value != rhs.value; }
    friend constexpr const ModInt operator+(const ModInt& lhs, const ModInt& rhs) noexcept { return ModInt(lhs) += rhs; }
    friend constexpr const ModInt operator-(const ModInt& lhs, const ModInt& rhs) noexcept { return ModInt(lhs) -= rhs; }
    friend constexpr const ModInt operator*(const ModInt& lhs, const ModInt& rhs) noexcept { return ModInt(lhs) *= rhs; }
    friend constexpr const ModInt operator/(const ModInt& lhs, const ModInt& rhs) noexcept { return ModInt(lhs) /= rhs; }

    friend std::ostream& operator<<(std::ostream& os, const ModInt& x) {
#ifdef LOCAL_DEBUG
        if (x.isNull()) return os << "{nullopt}";
#endif
        return os << x.value;
    }

    friend std::istream& operator>>(std::istream& is, ModInt& x) {
        is >> x.value;
        x.value = ModInt::normalized(x.value);
        return is;
    }

private:
    static constexpr int64_t normalized(int64_t n) {
        n = (-mod() <= n && n < mod() ? n : n % mod());
        if (n < 0) n += mod();
        return n;
    }

    static constexpr int64_t inverse(int64_t a, int64_t m) {
        int64_t u = 0, v = 1;
        while (a != 0) {
            const auto t = m / a;
            static_cast<void>(m -= t * a), std::swap(m, a);
            static_cast<void>(u -= t * v), std::swap(u, v);
        }
        assert(m == 1);
        return u;
    }
};

template <int64_t Mod>
struct StaticModHolder {
    static constexpr int64_t mod = Mod;
};

template <int ID>
struct DynamicModHolder {
    static int64_t mod;
};
template <int ID>
int64_t DynamicModHolder<ID>::mod;

}  // namespace internal

template <int64_t Mod>
using StaticModInt = internal::ModInt<internal::StaticModHolder<Mod>>;

using ModInt1000000007 = StaticModInt<int(1e9) + 7>;
using ModInt998244353 = StaticModInt<998244353>;

template <int ID>
using DynamicModInt = internal::ModInt<internal::DynamicModHolder<ID>>;
#line 6 "Math/Combinatorics/factorials.hpp"

/**
 * @brief factorials (階乗, 階乗の逆元, nCr, nPr)
 */
template <class Mint>
struct Factorials {
public:
    using value_type = Mint;
    static constexpr size_t MAX_N = std::min<size_t>(1e7, Mint::mod()) + 1;

private:
    mutable std::vector<value_type> m_fact, m_finv;

public:
    Factorials() {
        m_fact.reserve(MAX_N);
        m_finv.reserve(MAX_N);
        m_fact.resize(2, value_type::raw(1));   // m_fact[0] = m_fact[1] = 1
        m_finv.resize(2, value_type::raw(1));   // m_finv[0] = m_finv[1] = 1
    }

    void preCalc(size_t n) const {
        if (n < m_fact.size()) return;
        const size_t l = m_fact.size();
        const size_t r = n + 1;
        m_fact.resize(r), m_finv.resize(r);
        for (size_t i = l; i < r; ++i) m_fact[i] = m_fact[i - 1] * i;
        m_finv[r - 1] = m_fact[r - 1].inv();
        for (size_t i = r - 1; i > l; --i) m_finv[i - 1] = m_finv[i] * i;
    }

    const value_type fact(int i) const { return preCalc(i), m_fact[i]; }
    const value_type finv(int i) const { return preCalc(i), m_finv[i]; }

    const value_type C(int n, int r) const {
        if (r < 0 || n < r) return value_type::raw(0);
        return preCalc(n), m_fact[n] * m_finv[r] * m_finv[n - r];
    }

    const value_type P(int n, int r) const {
        if (r < 0 || n < r) return value_type::raw(0);
        return preCalc(n), m_fact[n] * m_finv[n - r];
    }

    const value_type H(int n, int r) const {
        if (n < 0 || r < 0) return value_type::raw(0);
        if (n == 0 && r == 0) return value_type::raw(1);
        return C(n + r - 1, r);
    }
};