noya2_Library

This documentation is automatically generated by online-judge-tools/verification-helper

View the Project on GitHub noya2ruler/noya2_Library

:heavy_check_mark: fps998244353/fps998244353.hpp

Depends on

Required by

Verified with

Code

#pragma once

#include"../utility/modint.hpp"

#include"ntt998244353.hpp"
#include"../math/binomial.hpp"

namespace noya2 {

// Formal Power Series for modint998244353
struct fps998244353 : std::vector<modint998244353> {
    using mint = modint998244353;
    using std::vector<mint>::vector;
    using std::vector<mint>::operator=;
    using fps = fps998244353;
    static inline binomial<mint> bnm;

    fps998244353 (const std::vector<mint> &init){
        (*this) = init;
    }

    void shrink(){
        while(!(this->empty()) && this->back().val() == 0){
            this->pop_back();
        }
    }

    fps &operator*= (const mint &r){
        for (auto &x : *this) x *= r;
        return *this;
    }
    fps &operator/= (const mint &r){
        (*this) *= r.inv();
        return *this;
    }

    fps &operator<<= (const int &d){
        this->insert(this->begin(), d, mint(0));
        return *this;
    }
    fps &operator>>= (const int &d){
        if ((int)(this->size()) <= d) this->clear();
        else this->erase(this->begin(),this->begin() + d);
        return *this;
    }

    fps &operator+= (const fps &r){
        if (this->size() < r.size()) this->resize(r.size());
        for (int i = 0; auto x : r){
            (*this)[i++] += x;
        }
        return *this;
    }
    fps &operator-= (const fps &r){
        if (this->size() < r.size()) this->resize(r.size());
        for (int i = 0; auto x : r){
            (*this)[i++] -= x;
        }
        return *this;
    }
    fps &operator*= (const fps &r){
        if (this->empty() || r.empty()){
            this->clear();
            return *this;
        }
        (*this) = ntt998244353::multiply(*this, r);
        return *this;
    }

    fps operator* (const mint &r) const { return fps(*this) *= r; }
    fps operator/ (const mint &r) const { return fps(*this) /= r; }
    fps operator<< (const int &d) const { return fps(*this) <<= d; }
    fps operator>> (const int &d) const { return fps(*this) >>= d; }

    fps operator+ (const fps &r) const { return fps(*this) += r; }
    fps operator- (const fps &r) const { return fps(*this) -= r; }
    fps operator* (const fps &r) const { return fps(*this) *= r; }

    fps operator+ () const { return *this; }
    fps operator- () const {
        fps ret(*this);
        for (auto &x : ret) x = -x;
        return ret;
    }

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

    [[nodiscard("Do not change but return changed object.")]]
    fps pre(std::size_t sz) const {
        fps ret(this->begin(), this->begin() + std::min(this->size(), sz));
        if (ret.size() < sz) ret.resize(sz);
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps rev() const {
        fps ret(*this);
        std::reverse(ret.begin(), ret.end());
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps diff() const {
        if (this->empty()){
            return fps();
        }
        fps ret(this->begin() + 1, this->end());
        for (int i = 1; auto &x : ret){
            x *= i++;
        }
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps integral() const {
        if (this->empty()){
            return fps();
        }
        fps ret(1, mint(0));
        ret.insert(ret.end(), this->begin(), this->end());
        for (int i = 0; auto &x : ret){
            x *= bnm.inv(i++); // inv(0) = 0
        }
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps inv(int d = -1) const {
        const int n = this->size();
        if (d == -1) d = n;
        fps res = {(*this)[0].inv()};
        for (int siz = 1; siz < d; siz <<= 1){
            fps f(this->begin(),this->begin()+min(n,siz*2)), g(res);
            f.resize(siz*2), g.resize(siz*2);
            f.ntt(), g.ntt();
            for (int i = 0; i < siz*2; i++) f[i] *= g[i];
            f.intt();
            f.erase(f.begin(),f.begin()+siz);
            f.resize(siz*2);
            f.ntt();
            for (int i = 0; i < siz*2; i++) f[i] *= g[i];
            f.intt();
            mint siz2_inv = mint(siz*2).inv(); siz2_inv *= -siz2_inv;
            for (int i = 0; i < siz; i++) f[i] *= siz2_inv;
            res.insert(res.end(),f.begin(),f.begin()+siz);
        }
        res.resize(d);
        return res;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps log(int d = -1) const {
        assert(this->empty() == false && (*this)[0].val() == 1u);
        if (d == -1) d = this->size();
        return (this->diff() * this->inv(d)).pre(d - 1).integral();
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps exp(int d = -1) const {
        const int n = this->size();
        if (d == -1) d = n;
        assert(n == 0 || (*this)[0].val() == 0u);
        if (n <= 1){
            fps ret(1,1);
            ret.resize(d);
            return ret;
        }
        // n >= 2
        fps f = {mint(1), (*this)[1]}, ret = f;
        for (int sz = 2; sz < d; sz <<= 1){
            f.insert(f.end(), this->begin()+std::min(n,sz), this->begin()+std::min(n,sz*2));
            f.resize(sz*2);
            ret *= f - ret.log(sz*2);
            ret.resize(sz*2);
        }
        ret.resize(d);
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps pow(long long k, int d = -1) const {
        const int n = this->size();
        if (d == -1) d = n;
        if (k == 0){
            fps ret(d, mint(0));
            if (d >= 1) ret[0] = 1;
            return ret;
        }
        // Find left-most nonzero term.
        for (int i = 0; i < n; i++){
            if ((*this)[i].val() != 0u){
                mint iv = (*this)[i].inv();
                fps ret = ((((*this) * iv) >> i).log(d) * mint(k)).exp(d);
                ret *= (*this)[i].pow(k);
                ret = (ret << (i * k)).pre(d);
                return ret;
            }
            if ((i + 1) * k >= d) break;
        }
        return fps(d, mint(0));
    }

    void ntt(){
        ntt998244353::ntt(*this);
    }
    // NOT /= len
    void intt(){
        ntt998244353::intt(*this);
    }
    // already /= len
    void intt_div(){
        ntt998244353::intt_div(*this);
    }
    //  input : ntt( f[0, 2^n) )
    // output : ntt( f[0, 2^n) ++ zero_padding[0, 2^n) )
    void ntt_doubling(){
        ntt998244353::ntt_doubling(*this);
    }
    //  input : ntt( f[0, 2^n) )
    // output : ntt( g[0, 2^{n-1}) ), g[i] = f[i * 2 + odd]
    void ntt_pick_parity(int odd){
        ntt998244353::ntt_pick_parity(*this, odd);
    }
    fps quotient(fps r) const {
        r.shrink();
        const int n = this->size(), m = r.size();
        if (n < m){
            return fps();
        }
        fps quo(*this);
        const int sz = n - m + 1;
        std::reverse(quo.begin(), quo.end());
        std::reverse(r.begin(), r.end());
        quo.resize(sz);
        quo *= r.inv(sz);
        quo.resize(sz);
        std::reverse(quo.begin(), quo.end());
        return quo;
    }
    fps remainder(fps r) const {
        r.shrink();
        const int n = this->size(), m = r.size();
        if (n < m){
            return fps(*this);
        }
        fps rem(*this);
        rem -= quotient(r) * r;
        rem.resize(m-1);
        rem.shrink();
        return rem;
    }
    std::pair<fps,fps> remquo(fps r) const {
        r.shrink();
        fps quo = quotient(r);
        fps rem(*this);
        rem -= quo * r;
        rem.shrink();
        return {rem, quo};
    }
};

} // namespace noya2
#line 2 "fps998244353/fps998244353.hpp"

#line 2 "utility/modint.hpp"

#include <iostream>

#line 2 "math/prime.hpp"

#include<utility>
namespace noya2 {

constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

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;
}

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_flag = is_prime_constexpr(n);

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};
    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; 
        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

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_flag = primitive_root_constexpr(m);

} // namespace noya2
#line 6 "utility/modint.hpp"

namespace noya2{

struct barrett {
    unsigned int _m;
    unsigned long long im;
    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
    unsigned int umod() const { return _m; }
    unsigned int mul(unsigned int a, unsigned int b) const {
        unsigned long long z = a;
        z *= b;
        unsigned long long x = (unsigned long long)((__uint128_t(z) * im) >> 64);
        unsigned int v = (unsigned int)(z - x * _m);
        if (_m <= v) v += _m;
        return v;
    }
};

template <int m>
struct static_modint {
    using mint = static_modint;
  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }
    constexpr static_modint() : _v(0) {}
    template<std::signed_integral T>
    constexpr static_modint(T v){
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template<std::unsigned_integral T>
    constexpr static_modint(T v){
        _v = (unsigned int)(v % umod());
    }
    constexpr 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;
    }
    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 = (uint)(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 = inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }
    friend constexpr mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend constexpr mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend constexpr mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend constexpr mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend constexpr bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend constexpr bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint& p) {
        return os << p.val();
    }
    friend std::istream &operator>>(std::istream &is, mint &a) {
        long long t; is >> t;
        a = mint(t);
        return (is);
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = is_prime_flag<m>;
};


template <int id> struct dynamic_modint {
    using mint = dynamic_modint;
  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template<std::signed_integral T>
    dynamic_modint(T v){
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template<std::unsigned_integral T>
    dynamic_modint(T v){
        _v = (unsigned int)(v % umod());
    }
    uint 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 = noya2::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 std::ostream &operator<<(std::ostream &os, const mint& p) {
        return os << p.val();
    }
    friend std::istream &operator>>(std::istream &is, mint &a) {
        long long t; is >> t;
        a = mint(t);
        return (is);
    }

  private:
    unsigned int _v;
    static barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> noya2::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

template<typename T>
concept Modint = requires (T &a){
    T::mod();
    a.inv();
    a.val();
    a.pow(declval<int>());
};

} // namespace noya2
#line 4 "fps998244353/fps998244353.hpp"

#line 2 "fps998244353/ntt998244353.hpp"

#line 2 "fps998244353/modint998244353.hpp"

#include <cassert>

#line 6 "fps998244353/modint998244353.hpp"

namespace noya2 {

template <>
struct static_modint<998244353> {
    using mint = static_modint;
  public:
    static constexpr int mod() { return 998244353; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }
    constexpr static_modint() : _v(0) {}
    template<std::signed_integral T>
    constexpr static_modint(T v){
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template<std::unsigned_integral T>
    constexpr static_modint(T v){
        _v = (unsigned int)(v % umod());
    }
    constexpr 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;
    }
    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 {
        assert(_v);
        return pow(umod() - 2);
    }
    friend constexpr mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend constexpr mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend constexpr mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend constexpr mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend constexpr bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend constexpr bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }
    friend std::ostream &operator<<(std::ostream &os, const mint& p) {
        return os << p.val();
    }
    friend std::istream &operator>>(std::istream &is, mint &a) {
        long long t; is >> t;
        a = mint(t);
        return (is);
    }

    unsigned int _v;
    static constexpr int primitive_root_constexpr_v = 3;
  private:
    static constexpr unsigned int umod() { return 998244353u; }
    static constexpr bool prime = true;
};

} // namespace noya2
#line 4 "fps998244353/ntt998244353.hpp"

#line 6 "fps998244353/ntt998244353.hpp"
#include <vector>

namespace noya2 {

namespace internal {

constexpr int FFT_MAX = 23;
constexpr unsigned FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 911660635U, 372528824U, 929031873U, 452798380U, 922799308U, 781712469U, 476477967U, 166035806U, 258648936U, 584193783U, 63912897U, 350007156U, 666702199U, 968855178U, 629671588U, 24514907U, 996173970U, 363395222U, 565042129U, 733596141U, 267099868U, 15311432U};
constexpr unsigned INV_FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 86583718U, 509520358U, 337190230U, 87557064U, 609441965U, 135236158U, 304459705U, 685443576U, 381598368U, 335559352U, 129292727U, 358024708U, 814576206U, 708402881U, 283043518U, 3707709U, 121392023U, 704923114U, 950391366U, 428961804U, 382752275U, 469870224U};
constexpr unsigned FFT_RATIOS[FFT_MAX] = {911660635U, 509520358U, 369330050U, 332049552U, 983190778U, 123842337U, 238493703U, 975955924U, 603855026U, 856644456U, 131300601U, 842657263U, 730768835U, 942482514U, 806263778U, 151565301U, 510815449U, 503497456U, 743006876U, 741047443U, 56250497U, 867605899U};
constexpr unsigned INV_FFT_RATIOS[FFT_MAX] = {86583718U, 372528824U, 373294451U, 645684063U, 112220581U, 692852209U, 155456985U, 797128860U, 90816748U, 860285882U, 927414960U, 354738543U, 109331171U, 293255632U, 535113200U, 308540755U, 121186627U, 608385704U, 438932459U, 359477183U, 824071951U, 103369235U};

} // namespace noya2::internal

struct ntt998244353 {
    using mint = modint998244353;
    static constexpr unsigned MO = modint998244353::mod();
    static constexpr unsigned MO2 = MO * 2;
    static void ntt(mint *as, int n){
        int m = n;
        if (m >>= 1){
            for (int i = 0; i < m; i++){
                const unsigned x = as[i + m]._v;
                as[i + m]._v = as[i]._v + MO - x;
                as[i]._v += x;
            }
        }
        if (m >>= 1){
            mint prod = 1;
            for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)){
                for (int i = i0; i < i0 + m; i++){
                    const unsigned x = (prod * as[i + m])._v;
                    as[i + m]._v = as[i]._v + MO - x;
                    as[i]._v += x;
                }
                prod *= mint::raw(internal::FFT_RATIOS[__builtin_ctz(++h)]);
            }
        }
        for (; m; ){
            if (m >>= 1){
                mint prod = 1;
                for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)){
                    for (int i = i0; i < i0 + m; i++){
                        const unsigned x = (prod * as[i + m])._v;
                        as[i + m]._v = as[i]._v + MO - x;
                        as[i]._v += x;
                    }
                    prod *= mint::raw(internal::FFT_RATIOS[__builtin_ctz(++h)]);
                }
            }
            if (m >>= 1){
                mint prod = 1;
                for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)){
                    for (int i = i0; i < i0 + m; i++){
                        const unsigned x = (prod * as[i + m])._v;
                        as[i]._v = (as[i]._v >= MO2 ? as[i]._v - MO2 : as[i]._v);
                        as[i + m]._v = as[i]._v + MO - x;
                        as[i]._v += x;
                    }
                    prod *= mint::raw(internal::FFT_RATIOS[__builtin_ctz(++h)]);
                }
            }
        }
        for (int i = 0; i < n; i++){
            as[i]._v = (as[i]._v >= MO2 ? as[i]._v - MO2 : as[i]._v);
            as[i]._v = (as[i]._v >= MO ? as[i]._v - MO : as[i]._v);
        }
    }
    static void intt(mint *as, int n){
        int m = 1;
        if (m < (n >> 1)){
            mint prod = 1;
            for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)){
                for (int i = i0; i < i0 + m; i++){
                    const unsigned long long y = as[i]._v + MO - as[i + m]._v;
                    as[i]._v += as[i + m]._v;
                    as[i + m]._v = prod._v * y % MO;
                }
                prod *= mint::raw(internal::INV_FFT_RATIOS[__builtin_ctz(++h)]);
            }
            m <<= 1;
        }
        for (; m < (n >> 1); m <<= 1){
            mint prod = 1;
            for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)){
                for (int i = i0; i < i0 + (m >> 1); i++){
                    const unsigned long long y = as[i]._v + MO2 - as[i + m]._v;
                    as[i]._v += as[i + m]._v;
                    as[i]._v = (as[i]._v >= MO2 ? as[i]._v - MO2 : as[i]._v);
                    as[i + m]._v = prod._v * y % MO;
                }
                for (int i = i0 + (m >> 1); i < i0 + m; i++){
                    const unsigned long long y = as[i]._v + MO - as[i + m]._v;
                    as[i]._v += as[i + m]._v;
                    as[i + m]._v = prod._v * y % MO;
                }
                prod *= mint::raw(internal::INV_FFT_RATIOS[__builtin_ctz(++h)]);
            }
        }
        if (m < n){
            for (int i = 0; i < m; i++){
                const unsigned y = as[i]._v + MO2 - as[i + m]._v;
                as[i]._v += as[i + m]._v;
                as[i + m]._v = y;
            }
        }
        for (int i = 0; i < n; i++){
            as[i]._v = (as[i]._v >= MO2 ? as[i]._v - MO2 : as[i]._v);
            as[i]._v = (as[i]._v >= MO ? as[i]._v - MO : as[i]._v);
        }
    }
    static void ntt(std::vector<mint> &as){
        ntt(as.data(), as.size());
    }
    static void intt(std::vector<mint> &as){
        intt(as.data(), as.size());
    }
    static void intt_div(std::vector<mint> &as){
        intt(as);
        int n = as.size();
        const mint inv_n = mint::raw(n).inv();
        for (int i = 0; i < n; i++){
            as[i] *= inv_n;
        }
    }
    static std::vector<mint> multiply(std::vector<mint> as, std::vector<mint> bs){
        if (as.empty() || bs.empty()) return {};
        const int len = as.size() + bs.size() - 1u;
        if (std::min(as.size(), bs.size()) <= 40u){
            std::vector<mint> s(len);
            for (int i = 0; i < (int)(as.size()); i++){
                for (int j = 0; j < (int)(bs.size()); j++){
                    s[i + j] += as[i] * bs[j];
                }
            }
            return s;
        }
        int n = 1;
        for (; n < len; n <<= 1) {}
        if (as.size() == bs.size() && as == bs){
            as.resize(n);
            ntt(as);
            for (int i = 0; i < n; i++){
                as[i] *= as[i];
            }
        }
        else {
            as.resize(n);
            ntt(as);
            bs.resize(n);
            ntt(bs);
            for (int i = 0; i < n; i++){
                as[i] *= bs[i];
            }
        }
        intt_div(as);
        as.resize(len);
        return as;
    }
    static void ntt_doubling(std::vector<mint> &as){
        auto bs = as;
        intt(bs);
        mint e = mint::raw(internal::FFT_ROOTS[std::countr_zero(as.size()) + 1]);
        mint iv = mint::raw(as.size()).inv();
        for (auto &x : bs){
            x *= iv;
            iv *= e;
        }
        ntt(bs);
        as.insert(as.end(), bs.begin(), bs.end());
    }
    static void ntt_pick_parity(std::vector<mint> &f, int odd){
        int n = f.size() / 2;
        mint i2 = mint::raw((mint::mod() + 1) >> 1);
        if (odd == 0){
            for (int i = 0; i < n; i++){
                f[i] = (f[i * 2] + f[i * 2 + 1]) * i2;
            }
            f.resize(n);
            return ;
        }
        mint ie = mint::raw(internal::INV_FFT_ROOTS[std::countr_zero(f.size())]);
        std::vector<mint> es = {i2};
        while ((int)(es.size()) != n){
            std::vector<mint> nes(es.size() * 2u);
            for (int i = 0; i < (int)(es.size()); i++){
                nes[i * 2 + 0] = es[i];
                nes[i * 2 + 1] = es[i] * ie;
            }
            ie *= ie;
            std::swap(es, nes);
        }
        for (int i = 0; i < n; i++){
            f[i] = (f[i * 2] - f[i * 2 + 1]) * es[i];
        }
        f.resize(n);
    }
};

} // namespace noya2
#line 2 "math/binomial.hpp"

#line 4 "math/binomial.hpp"
namespace noya2 {

template<typename mint>
struct binomial {
    binomial(int len = 300000){ extend(len); }
    static mint fact(int n){
        if (n < 0) return 0;
        while (n >= (int)_fact.size()) extend();
        return _fact[n];
    }
    static mint ifact(int n){
        if (n < 0) return 0;
        while (n >= (int)_fact.size()) extend();
        return _ifact[n];
    }
    static mint inv(int n){
        return ifact(n) * fact(n-1);
    }
    static mint C(int n, int r){
        if (!(0 <= r && r <= n)) return 0;
        return fact(n) * ifact(r) * ifact(n-r);
    }
    static mint P(int n, int r){
        if (!(0 <= r && r <= n)) return 0;
        return fact(n) * ifact(n-r);
    }
    static mint catalan(int n){
        return C(n * 2, n) * inv(n + 1);
    }
    inline mint operator()(int n, int r) { return C(n, r); }
    template<class... Cnts>
    static mint M(const Cnts&... cnts){
        return multinomial(0,1,cnts...);
    }
    static void initialize(int len = 2){
        _fact.clear();
        _ifact.clear();
        extend(len);
    }
  private:
    static mint multinomial(const int& sum, const mint& div_prod){
        if (sum < 0) return 0;
        return fact(sum) * div_prod;
    }
    template<class... Tail>
    static mint multinomial(const int& sum, const mint& div_prod, const int& n1, const Tail&... tail){
        if (n1 < 0) return 0;
        return multinomial(sum+n1,div_prod*ifact(n1),tail...);
    }
    static inline std::vector<mint> _fact, _ifact;
    static void extend(int len = -1){
        if (_fact.empty()){
            _fact = _ifact = {1,1};
        }
        int siz = _fact.size();
        if (len == -1) len = siz * 2;
        len = (int)min<long long>(len, mint::mod() - 1);
        if (len < siz) return ;
        _fact.resize(len+1), _ifact.resize(len+1);
        for (int i = siz; i <= len; i++) _fact[i] = _fact[i-1] * i;
        _ifact[len] = _fact[len].inv();
        for (int i = len; i > siz; i--) _ifact[i-1] = _ifact[i] * i;
    }
};

} // namespace noya2
#line 7 "fps998244353/fps998244353.hpp"

namespace noya2 {

// Formal Power Series for modint998244353
struct fps998244353 : std::vector<modint998244353> {
    using mint = modint998244353;
    using std::vector<mint>::vector;
    using std::vector<mint>::operator=;
    using fps = fps998244353;
    static inline binomial<mint> bnm;

    fps998244353 (const std::vector<mint> &init){
        (*this) = init;
    }

    void shrink(){
        while(!(this->empty()) && this->back().val() == 0){
            this->pop_back();
        }
    }

    fps &operator*= (const mint &r){
        for (auto &x : *this) x *= r;
        return *this;
    }
    fps &operator/= (const mint &r){
        (*this) *= r.inv();
        return *this;
    }

    fps &operator<<= (const int &d){
        this->insert(this->begin(), d, mint(0));
        return *this;
    }
    fps &operator>>= (const int &d){
        if ((int)(this->size()) <= d) this->clear();
        else this->erase(this->begin(),this->begin() + d);
        return *this;
    }

    fps &operator+= (const fps &r){
        if (this->size() < r.size()) this->resize(r.size());
        for (int i = 0; auto x : r){
            (*this)[i++] += x;
        }
        return *this;
    }
    fps &operator-= (const fps &r){
        if (this->size() < r.size()) this->resize(r.size());
        for (int i = 0; auto x : r){
            (*this)[i++] -= x;
        }
        return *this;
    }
    fps &operator*= (const fps &r){
        if (this->empty() || r.empty()){
            this->clear();
            return *this;
        }
        (*this) = ntt998244353::multiply(*this, r);
        return *this;
    }

    fps operator* (const mint &r) const { return fps(*this) *= r; }
    fps operator/ (const mint &r) const { return fps(*this) /= r; }
    fps operator<< (const int &d) const { return fps(*this) <<= d; }
    fps operator>> (const int &d) const { return fps(*this) >>= d; }

    fps operator+ (const fps &r) const { return fps(*this) += r; }
    fps operator- (const fps &r) const { return fps(*this) -= r; }
    fps operator* (const fps &r) const { return fps(*this) *= r; }

    fps operator+ () const { return *this; }
    fps operator- () const {
        fps ret(*this);
        for (auto &x : ret) x = -x;
        return ret;
    }

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

    [[nodiscard("Do not change but return changed object.")]]
    fps pre(std::size_t sz) const {
        fps ret(this->begin(), this->begin() + std::min(this->size(), sz));
        if (ret.size() < sz) ret.resize(sz);
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps rev() const {
        fps ret(*this);
        std::reverse(ret.begin(), ret.end());
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps diff() const {
        if (this->empty()){
            return fps();
        }
        fps ret(this->begin() + 1, this->end());
        for (int i = 1; auto &x : ret){
            x *= i++;
        }
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps integral() const {
        if (this->empty()){
            return fps();
        }
        fps ret(1, mint(0));
        ret.insert(ret.end(), this->begin(), this->end());
        for (int i = 0; auto &x : ret){
            x *= bnm.inv(i++); // inv(0) = 0
        }
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps inv(int d = -1) const {
        const int n = this->size();
        if (d == -1) d = n;
        fps res = {(*this)[0].inv()};
        for (int siz = 1; siz < d; siz <<= 1){
            fps f(this->begin(),this->begin()+min(n,siz*2)), g(res);
            f.resize(siz*2), g.resize(siz*2);
            f.ntt(), g.ntt();
            for (int i = 0; i < siz*2; i++) f[i] *= g[i];
            f.intt();
            f.erase(f.begin(),f.begin()+siz);
            f.resize(siz*2);
            f.ntt();
            for (int i = 0; i < siz*2; i++) f[i] *= g[i];
            f.intt();
            mint siz2_inv = mint(siz*2).inv(); siz2_inv *= -siz2_inv;
            for (int i = 0; i < siz; i++) f[i] *= siz2_inv;
            res.insert(res.end(),f.begin(),f.begin()+siz);
        }
        res.resize(d);
        return res;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps log(int d = -1) const {
        assert(this->empty() == false && (*this)[0].val() == 1u);
        if (d == -1) d = this->size();
        return (this->diff() * this->inv(d)).pre(d - 1).integral();
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps exp(int d = -1) const {
        const int n = this->size();
        if (d == -1) d = n;
        assert(n == 0 || (*this)[0].val() == 0u);
        if (n <= 1){
            fps ret(1,1);
            ret.resize(d);
            return ret;
        }
        // n >= 2
        fps f = {mint(1), (*this)[1]}, ret = f;
        for (int sz = 2; sz < d; sz <<= 1){
            f.insert(f.end(), this->begin()+std::min(n,sz), this->begin()+std::min(n,sz*2));
            f.resize(sz*2);
            ret *= f - ret.log(sz*2);
            ret.resize(sz*2);
        }
        ret.resize(d);
        return ret;
    }
    [[nodiscard("Do not change but return changed object.")]]
    fps pow(long long k, int d = -1) const {
        const int n = this->size();
        if (d == -1) d = n;
        if (k == 0){
            fps ret(d, mint(0));
            if (d >= 1) ret[0] = 1;
            return ret;
        }
        // Find left-most nonzero term.
        for (int i = 0; i < n; i++){
            if ((*this)[i].val() != 0u){
                mint iv = (*this)[i].inv();
                fps ret = ((((*this) * iv) >> i).log(d) * mint(k)).exp(d);
                ret *= (*this)[i].pow(k);
                ret = (ret << (i * k)).pre(d);
                return ret;
            }
            if ((i + 1) * k >= d) break;
        }
        return fps(d, mint(0));
    }

    void ntt(){
        ntt998244353::ntt(*this);
    }
    // NOT /= len
    void intt(){
        ntt998244353::intt(*this);
    }
    // already /= len
    void intt_div(){
        ntt998244353::intt_div(*this);
    }
    //  input : ntt( f[0, 2^n) )
    // output : ntt( f[0, 2^n) ++ zero_padding[0, 2^n) )
    void ntt_doubling(){
        ntt998244353::ntt_doubling(*this);
    }
    //  input : ntt( f[0, 2^n) )
    // output : ntt( g[0, 2^{n-1}) ), g[i] = f[i * 2 + odd]
    void ntt_pick_parity(int odd){
        ntt998244353::ntt_pick_parity(*this, odd);
    }
    fps quotient(fps r) const {
        r.shrink();
        const int n = this->size(), m = r.size();
        if (n < m){
            return fps();
        }
        fps quo(*this);
        const int sz = n - m + 1;
        std::reverse(quo.begin(), quo.end());
        std::reverse(r.begin(), r.end());
        quo.resize(sz);
        quo *= r.inv(sz);
        quo.resize(sz);
        std::reverse(quo.begin(), quo.end());
        return quo;
    }
    fps remainder(fps r) const {
        r.shrink();
        const int n = this->size(), m = r.size();
        if (n < m){
            return fps(*this);
        }
        fps rem(*this);
        rem -= quotient(r) * r;
        rem.resize(m-1);
        rem.shrink();
        return rem;
    }
    std::pair<fps,fps> remquo(fps r) const {
        r.shrink();
        fps quo = quotient(r);
        fps rem(*this);
        rem -= quo * r;
        rem.shrink();
        return {rem, quo};
    }
};

} // namespace noya2
Back to top page