test/src/math/sum_i^d_r^i/sum_of_exponential_times_polynomial.test.cpp

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• Last update: 2024-01-30 20:59:02+09:00
• Problem: https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial

Depends on

• 階乗テーブル (library/math/factorial.hpp)
• 冪乗テーブル (library/math/pow_mods.hpp)
• Product Of Differences (library/math/product_of_differences.hpp)
• $\displaystyle \sum _ i i ^ d r ^ i$ (library/math/sum_i^d_r^i.hpp)
• 線形篩 (library/number/linear_sieve.hpp)
• ラグランジュ補間 (library/polynomial/lagrange_interpolation.hpp)
• Multi Point Evaluation (library/polynomial/multi_point_eval.hpp)
• Shift of Sampling Points of Polynomial (library/polynomial/shift_of_sampling_points.hpp)
• Powers (library/sequence/powers.hpp)

Code

#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial"

#include <iostream>
#include <atcoder/modint>

#include "library/math/sum_i^d_r^i.hpp"

int main() {
    using mint = atcoder::modint998244353;
    int r, d;
    long long n;
    std::cin >> r >> d >> n;
    std::cout << suisen::sum_i_i_pow_d_r_pow_i<mint>(d, r).sum(n).val() << std::endl;
    return 0;
}

#line 1 "test/src/math/sum_i^d_r^i/sum_of_exponential_times_polynomial.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial"

#include <iostream>
#include <atcoder/modint>

#line 1 "library/math/sum_i^d_r^i.hpp"



#line 1 "library/sequence/powers.hpp"



#include <cstdint>
#line 1 "library/number/linear_sieve.hpp"



#include <cassert>
#include <numeric>
#include <vector>

namespace suisen {
// referece: https://37zigen.com/linear-sieve/
class LinearSieve {
    public:
        LinearSieve(const int n) : _n(n), min_prime_factor(std::vector<int>(n + 1)) {
            std::iota(min_prime_factor.begin(), min_prime_factor.end(), 0);
            prime_list.reserve(_n / 20);
            for (int d = 2; d <= _n; ++d) {
                if (min_prime_factor[d] == d) prime_list.push_back(d);
                const int prime_max = std::min(min_prime_factor[d], _n / d);
                for (int prime : prime_list) {
                    if (prime > prime_max) break;
                    min_prime_factor[prime * d] = prime;
                }
            }
        }
        int prime_num() const noexcept { return prime_list.size(); }
        /**
         * Returns a vector of primes in [0, n].
         * It is guaranteed that the returned vector is sorted in ascending order.
         */
        const std::vector<int>& get_prime_list() const noexcept  {
            return prime_list;
        }
        const std::vector<int>& get_min_prime_factor() const noexcept { return min_prime_factor; }
        /**
         * Returns a vector of `{ prime, index }`.
         * It is guaranteed that the returned vector is sorted in ascending order.
         */
        std::vector<std::pair<int, int>> factorize(int n) const noexcept {
            assert(0 < n and n <= _n);
            std::vector<std::pair<int, int>> prime_powers;
            while (n > 1) {
                int p = min_prime_factor[n], c = 0;
                do { n /= p, ++c; } while (n % p == 0);
                prime_powers.emplace_back(p, c);
            }
            return prime_powers;
        }
    private:
        const int _n;
        std::vector<int> min_prime_factor;
        std::vector<int> prime_list;
};
} // namespace suisen


#line 6 "library/sequence/powers.hpp"

namespace suisen {
    // returns { 0^k, 1^k, ..., n^k }
    template <typename mint>
    std::vector<mint> powers(uint32_t n, uint64_t k) {
        const auto mpf = LinearSieve(n).get_min_prime_factor();
        std::vector<mint> res(n + 1);
        res[0] = k == 0;
        for (uint32_t i = 1; i <= n; ++i) res[i] = i == 1 ? 1 : uint32_t(mpf[i]) == i ? mint(i).pow(k) : res[mpf[i]] * res[i / mpf[i]];
        return res;
    }
} // namespace suisen


#line 1 "library/math/factorial.hpp"



#line 6 "library/math/factorial.hpp"

namespace suisen {
    template <typename T, typename U = T>
    struct factorial {
        factorial() = default;
        factorial(int n) { ensure(n); }

        static void ensure(const int n) {
            int sz = _fac.size();
            if (n + 1 <= sz) return;
            int new_size = std::max(n + 1, sz * 2);
            _fac.resize(new_size), _fac_inv.resize(new_size);
            for (int i = sz; i < new_size; ++i) _fac[i] = _fac[i - 1] * i;
            _fac_inv[new_size - 1] = U(1) / _fac[new_size - 1];
            for (int i = new_size - 1; i > sz; --i) _fac_inv[i - 1] = _fac_inv[i] * i;
        }

        T fac(const int i) {
            ensure(i);
            return _fac[i];
        }
        T operator()(int i) {
            return fac(i);
        }
        U fac_inv(const int i) {
            ensure(i);
            return _fac_inv[i];
        }
        U binom(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[r] * _fac_inv[n - r];
        }
        template <typename ...Ds, std::enable_if_t<std::conjunction_v<std::is_integral<Ds>...>, std::nullptr_t> = nullptr>
        U polynom(const int n, const Ds& ...ds) {
            if (n < 0) return 0;
            ensure(n);
            int sumd = 0;
            U res = _fac[n];
            for (int d : { ds... }) {
                if (d < 0 or d > n) return 0;
                sumd += d;
                res *= _fac_inv[d];
            }
            if (sumd > n) return 0;
            res *= _fac_inv[n - sumd];
            return res;
        }
        U perm(const int n, const int r) {
            if (n < 0 or r < 0 or n < r) return 0;
            ensure(n);
            return _fac[n] * _fac_inv[n - r];
        }
    private:
        static std::vector<T> _fac;
        static std::vector<U> _fac_inv;
    };
    template <typename T, typename U>
    std::vector<T> factorial<T, U>::_fac{ 1 };
    template <typename T, typename U>
    std::vector<U> factorial<T, U>::_fac_inv{ 1 };
} // namespace suisen


#line 1 "library/math/pow_mods.hpp"



#line 5 "library/math/pow_mods.hpp"

namespace suisen {
    template <int base_as_int, typename mint>
    struct static_pow_mods {
        static_pow_mods() = default;
        static_pow_mods(int n) { ensure(n); }
        const mint& operator[](int i) const {
            ensure(i);
            return pows[i];
        }
        static void ensure(int n) {
            int sz = pows.size();
            if (sz > n) return;
            pows.resize(n + 1);
            for (int i = sz; i <= n; ++i) pows[i] = base * pows[i - 1];
        }
    private:
        static inline std::vector<mint> pows { 1 };
        static inline mint base = base_as_int;
        static constexpr int mod = mint::mod();
    };

    template <typename mint>
    struct pow_mods {
        pow_mods() = default;
        pow_mods(mint base, int n) : base(base) { ensure(n); }
        const mint& operator[](int i) const {
            ensure(i);
            return pows[i];
        }
        void ensure(int n) const {
            int sz = pows.size();
            if (sz > n) return;
            pows.resize(n + 1);
            for (int i = sz; i <= n; ++i) pows[i] = base * pows[i - 1];
        }
    private:
        mutable std::vector<mint> pows { 1 };
        mint base;
        static constexpr int mod = mint::mod();
    };
}


#line 1 "library/polynomial/lagrange_interpolation.hpp"



#line 1 "library/math/product_of_differences.hpp"



#include <deque>
#line 1 "library/polynomial/multi_point_eval.hpp"



#line 5 "library/polynomial/multi_point_eval.hpp"

namespace suisen {
    template <typename FPSType, typename T>
    std::vector<typename FPSType::value_type> multi_point_eval(const FPSType& f, const std::vector<T>& xs) {
        int n = xs.size();
        if (n == 0) return {};
        std::vector<FPSType> seg(2 * n);
        for (int i = 0; i < n; ++i) seg[n + i] = FPSType{ -xs[i], 1 };
        for (int i = n - 1; i > 0; --i) seg[i] = seg[i * 2] * seg[i * 2 + 1];
        seg[1] = f % seg[1];
        for (int i = 2; i < 2 * n; ++i) seg[i] = seg[i / 2] % seg[i];
        std::vector<typename FPSType::value_type> ys(n);
        for (int i = 0; i < n; ++i) ys[i] = seg[n + i].size() ? seg[n + i][0] : 0;
        return ys;
    }
} // namespace suisen


#line 6 "library/math/product_of_differences.hpp"

namespace suisen {
    /**
     * O(N(logN)^2)
     * return the vector p of length xs.size() s.t. p[i]=Π[j!=i](x[i]-x[j])
     */
    template <typename FPSType, typename T>
    std::vector<typename FPSType::value_type> product_of_differences(const std::vector<T>& xs) {
        // f(x):=Π_i(x-x[i])
        // => f'(x)=Σ_i Π[j!=i](x-x[j])
        // => f'(x[i])=Π[j!=i](x[i]-x[j])
        const int n = xs.size();
        std::deque<FPSType> dq;
        for (int i = 0; i < n; ++i) dq.push_back(FPSType{ -xs[i], 1 });
        while (dq.size() >= 2) {
            auto f = std::move(dq.front());
            dq.pop_front();
            auto g = std::move(dq.front());
            dq.pop_front();
            dq.push_back(f * g);
        }
        auto f = std::move(dq.front());
        f.diff_inplace();
        return multi_point_eval<FPSType, T>(f, xs);
    }
} // namespace suisen



#line 5 "library/polynomial/lagrange_interpolation.hpp"

namespace suisen {
    // O(N^2+NlogP)
    template <typename T>
    T lagrange_interpolation_naive(const std::vector<T>& xs, const std::vector<T>& ys, const T t) {
        const int n = xs.size();
        assert(int(ys.size()) == n);

        T p{ 1 };
        for (int i = 0; i < n; ++i) p *= t - xs[i];

        T res{ 0 };
        for (int i = 0; i < n; ++i) {
            T w = 1;
            for (int j = 0; j < n; ++j) if (j != i) w *= xs[i] - xs[j];
            res += ys[i] * (t == xs[i] ? 1 : p / (w * (t - xs[i])));
        }
        return res;
    }

    // O(N(logN)^2+NlogP)
    template <typename FPSType, typename T>
    typename FPSType::value_type lagrange_interpolation(const std::vector<T>& xs, const std::vector<T>& ys, const T t) {
        const int n = xs.size();
        assert(int(ys.size()) == n);

        std::vector<FPSType> seg(2 * n);
        for (int i = 0; i < n; ++i) seg[n + i] = FPSType {-xs[i], 1};
        for (int i = n - 1; i > 0; --i) seg[i] = seg[i * 2] * seg[i * 2 + 1];
        seg[1] = seg[1].diff() % seg[1];
        for (int i = 2; i < 2 * n; ++i) seg[i] = seg[i / 2] % seg[i];

        using mint = typename FPSType::value_type;
        mint p{ 1 };
        for (int i = 0; i < n; ++i) p *= t - xs[i];

        mint res{ 0 };
        for (int i = 0; i < n; ++i) {
            mint w = seg[n + i][0];
            res += ys[i] * (t == xs[i] ? 1 : p / (w * (t - xs[i])));
        }
        return res;
    }

    // xs[i] = ai + b
    // requirement: for all 0≤i<j<n, ai+b ≢ aj+b mod p
    template <typename T>
    T lagrange_interpolation_arithmetic_progression(T a, T b, const std::vector<T>& ys, const T t) {
        const int n = ys.size();
        T fac = 1;
        for (int i = 1; i < n; ++i) fac *= i;
        std::vector<T> fac_inv(n), suf(n);
        fac_inv[n - 1] = T(1) / fac;
        suf[n - 1] = 1;
        for (int i = n - 1; i > 0; --i) {
            fac_inv[i - 1] = fac_inv[i] * i;
            suf[i - 1] = suf[i] * (t - (a * i + b));
        }
        T pre = 1, res = 0;
        for (int i = 0; i < n; ++i) {
            T val = ys[i] * pre * suf[i] * fac_inv[i] * fac_inv[n - i - 1];
            if ((n - 1 - i) & 1) res -= val;
            else                 res += val;
            pre *= t - (a * i + b);
        }
        return res / a.pow(n - 1);
    }
    // x = 0, 1, ...
    template <typename T>
    T lagrange_interpolation_arithmetic_progression(const std::vector<T>& ys, const T t) {
        return lagrange_interpolation_arithmetic_progression(T{1}, T{0}, ys, t);
    }
} // namespace suisen



#line 1 "library/polynomial/shift_of_sampling_points.hpp"



#line 5 "library/polynomial/shift_of_sampling_points.hpp"
#include <atcoder/convolution>

#line 8 "library/polynomial/shift_of_sampling_points.hpp"

namespace suisen {
    template <typename mint, typename Convolve,
        std::enable_if_t<std::is_invocable_r_v<std::vector<mint>, Convolve, std::vector<mint>, std::vector<mint>>, std::nullptr_t> = nullptr>
    std::vector<mint> shift_of_sampling_points(const std::vector<mint>& ys, mint t, int m, const Convolve &convolve) {
        const int n = ys.size();
        factorial<mint> fac(std::max(n, m));

        std::vector<mint> b = [&] {
            std::vector<mint> f(n), g(n);
            for (int i = 0; i < n; ++i) {
                f[i] = ys[i] * fac.fac_inv(i);
                g[i] = (i & 1 ? -1 : 1) * fac.fac_inv(i);
            }
            std::vector<mint> b = convolve(f, g);
            b.resize(n);
            return b;
        }();
        std::vector<mint> e = [&] {
            std::vector<mint> c(n);
            mint prd = 1;
            std::reverse(b.begin(), b.end());
            for (int i = 0; i < n; ++i) {
                b[i] *= fac.fac(n - i - 1);
                c[i] = prd * fac.fac_inv(i);
                prd *= t - i;
            }
            std::vector<mint> e = convolve(b, c);
            e.resize(n);
            return e;
        }();
        std::reverse(e.begin(), e.end());
        for (int i = 0; i < n; ++i) {
            e[i] *= fac.fac_inv(i);
        }

        std::vector<mint> f(m);
        for (int i = 0; i < m; ++i) f[i] = fac.fac_inv(i);
        std::vector<mint> res = convolve(e, f);
        res.resize(m);
        for (int i = 0; i < m; ++i) res[i] *= fac.fac(i);
        return res;
    }

    template <typename mint>
    std::vector<mint> shift_of_sampling_points(const std::vector<mint>& ys, mint t, int m) {
        auto convolve = [&](const std::vector<mint> &f, const std::vector<mint> &g) { return atcoder::convolution(f, g); };
        return shift_of_sampling_points(ys, t, m, convolve);
    }
} // namespace suisen



#line 9 "library/math/sum_i^d_r^i.hpp"

namespace suisen {
    template <typename mint>
    struct sum_i_i_pow_d_r_pow_i {
        sum_i_i_pow_d_r_pow_i(int d, mint r) : d(d), r(r), i_pow_d(powers<mint>(d + 1, d)), r_pow_i(r, d + 1), fac(d), c(calc_c()) {}

        mint sum() const {
            assert(r != 1);
            return c;
        }
        mint sum(long long n) {
            if (r == 0) return n > 0 and d == 0 ? 1 : 0;
            prepare();
            return lagrange_interpolation_arithmetic_progression<mint>(ys, n) * r.pow(n) + c;
        }
        std::vector<mint> sum(long long t, int m) {
            if (r == 0) {
                std::vector<mint> res(m);
                for (long long n = t; n < t + m; ++n) res[n - t] = sum(n);
                return res;
            }
            prepare();
            auto res = shift_of_sampling_points<mint>(ys, t, m);
            mint pr = r.pow(r);
            for (auto &e : res) e *= pr, e += c, pr *= r;
            return res;
        }
    private:
        int d;
        mint r;
        std::vector<mint> i_pow_d;
        pow_mods<mint> r_pow_i;
        factorial<mint> fac;

        mint c;

        std::vector<mint> ys;
        bool prepared = false;

        mint calc_c() {
            if (r == 1) return 0;
            mint num = 0, den = 0, sum = 0;
            for (int i = 0; i <= d + 1; ++i) {
                sum += i_pow_d[i] * r_pow_i[i];
                den += (i & 1 ? -1 : +1) * fac.binom(d + 1, i) * r_pow_i[i];
                num += ((d + 1 - i) & 1 ? -1 : +1) * fac.binom(d + 1, d + 1 - i) * r_pow_i[d + 1 - i] * sum;
            }
            return num / den;
        }

        void prepare() {
            if (prepared) return;
            prepared = true;

            ys.resize(d + 2);
            for (int i = 0; i <= d; ++i) ys[i + 1] = ys[i] + r_pow_i[i] * i_pow_d[i];
            
            if (r == 1) return;

            for (auto& e : ys) e -= c;

            mint inv_r = r.inv();
            mint pow_inv_r = inv_r.pow(d + 1);
            for (int i = d + 1; i >= 0; --i) {
                ys[i] *= pow_inv_r;
                pow_inv_r *= r;
            }
        }
    };
} // namespace suisen



#line 7 "test/src/math/sum_i^d_r^i/sum_of_exponential_times_polynomial.test.cpp"

int main() {
    using mint = atcoder::modint998244353;
    int r, d;
    long long n;
    std::cin >> r >> d >> n;
    std::cout << suisen::sum_i_i_pow_d_r_pow_i<mint>(d, r).sum(n).val() << std::endl;
    return 0;
}

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