#1172617 (C++23) No.3771 Leq-K Partition Multiplied

• 問題: No.3771 Leq-K Partition Multiplied
• ユーザー: apricity
• 提出日時: 2026-07-23 16:04:36
• 言語: C++23 (gcc 15.3.0 + boost 1.92.0 + ACL)
• 結果: AC
• 実行時間: 2,598 ms / 8,000 ms
• コード長: 16,160 bytes
• コンパイル時間: 3,965 ms
• コンパイル使用メモリ: 289,708 KB
• 実行使用メモリ: 289,060 KB
• 最終ジャッジ日時: 2026-10-16 20:54:19
• 合計ジャッジ時間: 27,114 ms
• ジャッジサーバーID （参考情報）: judge1_0 / judge5_0

ファイルパターン	結果
sample	AC * 3
other	AC * 12

ソースコード

#include <iostream>
#include <vector>
#include <queue>
#include <random>
using namespace std;
#include "atcoder/modint.hpp"
#include "atcoder/convolution.hpp"

template <typename mint>
struct Polynomial : vector<mint> {
    using vector<mint>::vector;
    using poly = Polynomial<mint>;

    void shrink() {
        while(!this->empty() and this->back() == mint(0)) this->pop_back();
    }

    poly rev() const {
        poly res(*this);
        reverse(res.begin(), res.end());
        return res;
    }

    poly pre(int sz) const {
        poly res(this->begin(), this->begin() + min(this->size(), (size_t)sz));
        if (res.size() < (size_t)sz) res.resize(sz);
        return res;
    }

    poly &operator>>=(int sz) {
        if (this->size() <= (size_t)sz) {
            this->clear();
            return *this;
        }
        this->erase(this->begin(), this->begin() + (size_t)sz);
        return *this;
    }

    poly &operator<<=(int sz) {
        this->insert(this->begin(), sz, mint::raw(0));
        return *this;
    }

    poly &operator+=(const mint &s) {
        if (this->empty()) this->resize(1);
        (*this)[0] += s;
        return *this;
    }

    poly &operator+=(const poly &p) {
        if (this->size() < p.size()) this->resize(p.size());
        for (size_t i = 0; i < p.size(); i++) (*this)[i] += p[i];
        return *this;
    }

    poly &operator-=(const mint &s) {
        if (this->empty()) this->resize(1);
        (*this)[0] -= s;
        return *this;
    }

    poly &operator-=(const poly &p) {
        if (this->size() < p.size()) this->resize(p.size());
        for (size_t i = 0; i < p.size(); i++) (*this)[i] -= p[i];
        return *this;
    }

    poly &operator*=(const mint &s) {
        for (size_t i = 0; i < this->size(); i++) (*this)[i] *= s;
        return *this;
    }

    poly operator*=(const poly &p) {
        if (this->empty() or p.empty()) {
            this->clear();
            return *this;
        }
        vector<mint> prod = atcoder::convolution(*this, p);
        return *this = poly(prod.begin(), prod.end());
    }

    poly &operator/=(const mint &s) {
        mint s_inv = s.inv();
        for (size_t i = 0; i < this->size(); i++) (*this)[i] *= s_inv;
        return *this;
    }

    poly operator/=(const poly &p) {
        if (this->size() < p.size()) {
            this->clear();
            return *this;
        }
        size_t q_sz = this->size() - p.size() + 1;
        return *this = (this->rev().pre(q_sz) * p.rev().inv(q_sz)).pre(q_sz).rev();
    }

    poly operator%=(const poly &p) {
        *this -= (*this / p) * p;
        shrink();
        assert(this->size() < p.size());
        return *this;
    }

    poly operator+(const mint &s) const { return poly(*this) += s; }
    poly operator+(const poly &p) const { return poly(*this) += p; }
    poly operator-(const mint &s) const { return poly(*this) -= s; }
    poly operator-(const poly &p) const { return poly(*this) -= p; }
    poly operator*(const mint &s) const { return poly(*this) *= s; }
    poly operator*(const poly &p) const { return poly(*this) *= p; }
    poly operator/(const mint &s) const { return poly(*this) /= s; }
    poly operator/(const poly &p) const { return poly(*this) /= p; }
    poly operator%(const poly &p) const { return poly(*this) %= p; }
    poly operator>>(int s) const { return poly(*this) >>= s; }
    poly operator<<(int s) const { return poly(*this) <<= s; }
    poly operator-() const {
        poly res(this->size());
        for (size_t i = 0; i < this->size(); i++) res[i] = -(*this)[i];
        return res;
    }

    poly dot(const poly &p) const {
        poly res(min(this->size(), p.size()));
        for (size_t i = 0; i < res.size(); i++) res[i] = (*this)[i] * p[i];
        return res;
    }

    mint eval(mint x) const {
        mint res = 0, pw = 1;
        for (auto &c : *this) res += c * pw, pw *= x;
        return res;
    }

    void ntt() {
        atcoder::internal::butterfly(*this);
    }

    void intt() {
        atcoder::internal::butterfly_inv(*this);
        mint sz_inv = mint::raw(this->size()).inv();
        for (auto &c : *this) c *= sz_inv;
    }

    poly inv(int deg = -1) const {
        assert(!this->empty() and (*this)[0] != 0);
        if (deg == -1) deg = (int)this->size();
        poly res(deg);
        res[0] = (*this)[0].inv();
        for (int d = 1; d < deg; d <<= 1) {
            poly f(2 * d), g(2 * d);
            for (int j = 0; j < min((int)this->size(), 2 * d); j++) f[j] = (*this)[j];
            for (int j = 0; j < d; j++) g[j] = res[j];
            // atcoder::internal::butterfly(f);
            f.ntt();
            // atcoder::internal::butterfly(g);
            g.ntt();
            for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
            // atcoder::internal::butterfly_inv(f);
            f.intt();
            for (int j = 0; j < d; j++) f[j] = 0;
            // atcoder::internal::butterfly(f);
            f.ntt();
            for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
            // atcoder::internal::butterfly_inv(f);
            f.intt();
            for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
        }
        return res.pre(deg);
    }

    poly differentiate() const {
        const size_t n = this->size();
        poly res(n == 0 ? 0 : n - 1);
        mint coef = 1;
        for (size_t i = 1; i < n; i++) {
            res[i - 1] = (*this)[i] * coef;
            coef += 1;
        }
        return res;
    }

    poly integrate() const {
        const size_t n = this->size();
        poly res(n + 1);
        res[0] = mint::raw(0);
        if (n > 0) res[1] = mint(1);
        int mod = mint::mod();
        for (size_t i = 2; i <= n; i++) res[i] = (-res[mod % i]) * (mod / i);
        for (size_t i = 0; i < n; i++) res[i + 1] *= (*this)[i];
        return res;
    }

    poly log(int deg = -1) const {
        assert(!this->empty() and (*this)[0] == 1);
        if (deg == -1) deg = (int)this->size();
        return (this->differentiate() * this->inv(deg)).pre(deg - 1).integrate();
    }

    poly exp(int deg = -1) const {
        assert(this->empty() or (*this)[0] == mint(0));
        if (deg == -1) deg = (int)this->size();

        vector<mint> inv;
        inv.reserve(deg + 1);
        inv.emplace_back(mint::raw(0));
        inv.emplace_back(mint::raw(1));
        poly b{1, 1 < (int)this->size() ? (*this)[1] : 0};
        poly c{1}, z1, z2{1, 1};
        int mod = mint::mod();
        for (int d = 2; d < deg ; d <<= 1) {
            poly y = b;
            y.resize(2 * d);
            // atcoder::internal::butterfly(y);
            y.ntt();
            z1 = z2;
            poly z(d);
            for (int i = 0; i < d; i++) z[i] = y[i] * z1[i];
            // atcoder::internal::butterfly_inv(z);
            z.intt();
            fill(z.begin(), z.begin() + d / 2, mint::raw(0));
            // atcoder::internal::butterfly(z);
            z.ntt();
            for (int i = 0; i < d; i++) z[i] *= -z1[i];
            // atcoder::internal::butterfly_inv(z);
            z.intt();
            c.insert(c.end(), z.begin() + d / 2, z.end());
            z2 = c;
            z2.resize(2 * d);
            // atcoder::internal::butterfly(z2);
            z2.ntt();
            poly x(this->begin(), this->begin() + min<int>(this->size(), d));
            x.resize(d);
            {
                x.erase(x.begin());
                mint coef = 1;
                for (int i = 0; i < d-1; i++) x[i] *= coef, coef += 1;
            }
            x.emplace_back(mint::raw(0));
            // atcoder::internal::butterfly(x);
            x.ntt();
            for (int i = 0; i < d; i++) x[i] *= y[i];
            // atcoder::internal::butterfly_inv(x);
            x.intt();
            x -= b.differentiate();
            x.resize(2 * d);
            for (int i = 0; i < d - 1; i++) x[d + i] = x[i], x[i] = mint::raw(0);
            // atcoder::internal::butterfly(x);
            x.ntt();
            for (int i = 0; i < 2 * d; i++) x[i] *= z2[i];
            // atcoder::internal::butterfly_inv(x);
            x.intt();
            x.pop_back();
            {
                int sz;
                while((sz = (int)inv.size()) <= (int)x.size()) {
                    inv.emplace_back((-inv[mod % sz]) * (mod / sz));
                }
                x.insert(x.begin(), mint::raw(0));
                for (size_t i = 1; i < x.size(); i++) x[i] *= inv[i];
            }
            for (int i = d; i < min((int)this->size(), 2 * d); i++) x[i] += (*this)[i];
            fill(x.begin(), x.begin() + d, mint::raw(0));
            // atcoder::internal::butterfly(x);
            x.ntt();
            for (int i = 0; i < 2 * d; i++) x[i] *= y[i];
            // atcoder::internal::butterfly_inv(x);
            x.intt();
            b.insert(b.end(), x.begin() + d, x.end());
        }
        return b.pre(deg);
    }

    poly pow(long long k, int deg = -1) const {
        const int n = (int)this->size();
        if (deg == -1) deg = n;
        if (k == 0) {
            poly res(deg);
            if (deg > 0) res[0] = 1;
            return res;
        }
        int l = 0;
        while (l < n and (*this)[l] == 0){
            l++;
            if (l >= (deg + k - 1) / k) return poly(deg, mint::raw(0));
        }
        if (l == n) return poly(deg, mint::raw(0));
        mint rev = (*this)[l].inv();
        poly res = (((*this * rev) >> l).log(deg) * k).exp(deg);
        res *= (*this)[l].pow(k);
        res = (res << (k * l)).pre(deg);
        if ((int)res.size() < deg) res.resize(deg, mint::raw(0));
        return res;
    }
};

template<typename T> struct Binomial {
    vector<T> fact_, inv_, finv_;
    constexpr Binomial() {}
    constexpr Binomial(int n) noexcept : fact_(n, 1), inv_(n, 1), finv_(n, 1) {
        init(n);
    }
    constexpr void init(int n) noexcept {
        constexpr int mod = T::mod();
        fact_.assign(n, 1), inv_.assign(n, 1), finv_.assign(n, 1);
        for(int i = 2; i < n; i++){
            fact_[i] = fact_[i-1] * i;
            inv_[i] = -inv_[mod%i] * (mod/i);
            finv_[i] = finv_[i-1] * inv_[i];
        }
    }
    constexpr T com(int n, int k) const noexcept {
        if (n < k || n < 0 || k < 0) return 0;
        return fact_[n] * finv_[k] * finv_[n-k];
    }
    constexpr T perm(int n, int k) const noexcept {
        if (n < k || n < 0 || k < 0) return 0;
        return fact_[n] * finv_[n-k];
    }
    constexpr T fact(int n) const noexcept {
        if (n < 0) return 0;
        return fact_[n];
    }
    constexpr T inv(int n) const noexcept {
        if (n < 0) return 0;
        return inv_[n];
    }
    constexpr T finv(int n) const noexcept {
        if (n < 0) return 0;
        return finv_[n];
    }
    constexpr T com_naive(int n, int k) const noexcept {
        if (n < 0 || k < 0 || n < k) return 0;
        T res = T(1);
        k = min(k, n-k);
        for (int i = 1; i <= k; i++)res *= (n--) * inv(i);
        return res;
    }
    template <typename I>
    constexpr T multi(const vector<I> &v) const noexcept {
        static_assert(is_integral<I>::value);
        I n = 0;
        for (auto& x : v) {
            if (x < 0) return 0;
            n += x;
        }
        T res = fact(n);
        for (auto &x : v) res *= finv(x);
        return res;
    }
    // [x^k] (1-x)^{-n} = com(n+k-1, k)
    constexpr T neg(int n, int k) const noexcept {
        if (n < 0 || k < 0) return 0;
        return k == 0 ? 1 : com(n+k-1, k);
    }
};

using mint = atcoder::modint998244353;
using poly = Polynomial<mint>;
Binomial<mint> bc(300000);

// deg(f) = n-1
// [x^{n-1}] g(x)f(x)^i for i=0,1,...,m
template <typename mint>
Polynomial<mint> power_projection(Polynomial<mint> f, Polynomial<mint> g = {1}, int m = -1) {
    using poly = Polynomial<mint>;
    if (f.empty()) return poly(m+1,0);
    if (f[0] != 0) {
        mint c = f[0];
        f[0] = 0;
        poly a = power_projection(f, g, m);
        for (int i = 0; i <= m; i++) a[i] *= bc.finv(i);
        poly b(m+1);
        mint pc = 1;
        for (int i = 0; i <= m; i++) b[i] = bc.finv(i) * pc, pc *= c;
        a *= b; a.resize(m+1);
        for (int i = 0; i <= m; i++) a[i] *= bc.fact(i);
        return a;
    }

    int fn = f.size();
    if(m == -1) m = fn - 1;
    g.resize(fn);
    int n = 1;
    while (n < fn) n *= 2;
    f.resize(n);
    reverse(g.begin(), g.end());
    g.resize(n);
    reverse(g.begin(), g.end());

    int k = 1;
    poly p(n * 2), q(n * 2), r(n * 2);
    for (int i = 0; i < n; i++) p[i] = g[i], q[i] = -f[i];

    while (n > 1) {
        for (int i = 0; i < 2 * n * k; i++) r[i] = (i % 2 == 0 ? q[i] : -q[i]);
        poly pq = p * r;
        poly qq = q * r;
        pq.resize(4 * n * k);
        qq.resize(4 * n * k);
        for (int i = 0; i < 2 * n * k; i++) {
            pq[2 * n * k + i] += p[i];
            qq[2 * n * k + i] += q[i] + r[i];
        }
        fill(p.begin(), p.end(), 0);
        fill(q.begin(), q.end(), 0);
        for (int i = 0; i < 2 * k; i++) {
            for (int j = 0; j < n / 2; j++) {
                p[n * i + j] = pq[2 * n * i + 2 * j + 1];
                q[n * i + j] = qq[2 * n * i + 2 * j + 0];
            }
        }
        n /= 2; k *= 2;
    }
    poly pk(k),qk(k+1);
    for (int i = 0; i < k; i++) pk[i] = p[2 * i];
    for (int i = 0; i < k; i++) qk[i] = q[2 * i];
    qk[k] = 1;
    reverse(pk.begin(), pk.end());
    reverse(qk.begin(), qk.end());
    return (pk * qk.inv(m+1)).pre(m+1);
}

// g(f(x))
template <typename mint>
Polynomial<mint> composition(Polynomial<mint> f, Polynomial<mint> g, int deg = -1) {
    using poly = Polynomial<mint>;

    auto middle_product = [&] (poly x, poly y) -> poly {
        int nm_1 = x.size(), m = y.size();
        int l = bit_ceil(x.size());
        reverse(y.begin(), y.end());
        x.resize(l); y.resize(l);
        x.ntt(); y.ntt();
        for (int i = 0; i < l; i++) x[i] *= y[i];
        x.intt();
        return poly{x.begin() + m - 1, x.begin() + nm_1};
    };

    auto rec = [&] (this auto self, int n, int k, poly q) -> poly {
        if (n == 1) {
            reverse(g.begin(), g.end());
            poly p(2 * k);
            for (int i = 0; i < k; i++) p[2 * i] = g[i];
            return p;
        }
        poly r(2 * n * k);
        for (int i = 0; i < 2 * n * k; i++) r[i] = (i % 2 == 0 ? q[i] : -q[i]);
        poly qq = q * r;
        qq.resize(4 * n * k);
        for (int i = 0; i < 2 * n * k; i++) qq[2 * n * k + i] += q[i] + r[i];
        poly nq(2 * n * k);
        for (int i = 0; i < k * 2; i++) {
            for (int j = 0; j < n / 2; j++) {
                nq[n * i + j] = qq[2 * n * i + 2 * j];
            }
        }
        poly np = self(n / 2, k * 2, nq);
        poly pq(4 * n * k);
        for (int i = 0; i < k * 2; i++) {
            for (int j = 0; j < n / 2; j++) {
                pq[2 * n * i + 2 * j + 1] += np[n * i + j];
            }
        }
        poly p(2 * n * k);
        for (int i = 0; i < 2 * n * k; i++) p[i] += pq[2 * n * k + i];
        pq.pop_back();
        poly mp = middle_product(pq, r);
        for (int i = 0; i < 2 * n * k; i++) p[i] += mp[i];
        return p;
    };

    if (deg == -1) deg = max(f.size(), g.size());
    int k = 1, n = 1;
    while (n < deg) n *= 2;
    f.resize(n); g.resize(n);
    poly q(n * 2);
    for (int i = 0; i < n; i++) q[i] = -f[i];
    poly p = rec(n, k, q);
    return p.pre(n).rev().pre(deg);
}

int main() {
    long n,m,k,x; cin >> n >> m >> k >> x;
    poly f(n);
    for (int i = 1; i < n; i++) f[i] = bc.finv(i);
    f = f.pow(k, n);
    for (int i = 0; i < n; i++) f[i] *= bc.fact(i) * bc.finv(k);
    mint ff = 1;
    for (int i = 0; i < k; i++) ff *= m-1-i;
    for (int i = 0; i < n; i++) f[i] *= ff;
    poly c(n);
    for (int i = 0; i < n; i++) c[i] = mint(i+1).pow(x) - mint(i).pow(x);
    poly h = composition(f,c,n);
    vector<mint> ans(n);
    ans[0] = h[0];
    for (int i = 1; i < n; i++) ans[i] = ans[i-1] * m + h[i];
    for (int i = 0; i < n; i++) ans[i] *= m;
    for (int i = 0; i < n; i++) cout << ans[i].val() << "\n";
}