#1141294 (C++17) No.3414 Aperiodic Sequence

• 問題: No.3414 Aperiodic Sequence
• ユーザー: Nachia
• 提出日時: 2025-12-21 00:18:24
• 言語: C++17 (gcc 15.2.0 + boost 1.89.0)
• 結果: AC
• 実行時間: 427 ms / 3,000 ms
• コード長: 13,197 bytes
• コンパイル時間: 1,172 ms
• コンパイル使用メモリ: 94,252 KB
• 実行使用メモリ: 44,436 KB
• 最終ジャッジ日時: 2025-12-21 00:18:34
• 合計ジャッジ時間: 5,718 ms
• ジャッジサーバーID （参考情報）: judge4 / judge2

ファイルパターン	結果
sample	AC * 2
other	AC * 15

ソースコード

#ifdef NACHIA
#define _GLIBCXX_DEBUG
#else
#define NDEBUG
#endif
#include <iostream>
#include <string>
#include <vector>
#include <algorithm>
using namespace std;
using ll = long long;
const ll INF = 1ll << 60;
#define REP(i,n) for(ll i=0; i<ll(n); i++)
template <class T> using V = vector<T>;
template <class A, class B> void chmax(A& l, const B& r){ if(l < r) l = r; }
template <class A, class B> void chmin(A& l, const B& r){ if(r < l) l = r; }

#include <utility>

#include <cassert>
namespace nachia{

// ax + by = gcd(a,b)
// return ( x, - )
std::pair<long long, long long> ExtGcd(long long a, long long b){
    long long x = 1, y = 0;
    while(b){
        long long u = a / b;
        std::swap(a-=b*u, b);
        std::swap(x-=y*u, y);
    }
    return std::make_pair(x, a);
}

} // namespace nachia

namespace nachia{

template<unsigned int MOD>
struct StaticModint{
private:
    using u64 = unsigned long long;
    unsigned int x;
public:

    using my_type = StaticModint;
    template< class Elem >
    static Elem safe_mod(Elem x){
        if(x < 0){
            if(0 <= x+MOD) return x + MOD;
            return MOD - ((-(x+MOD)-1) % MOD + 1);
        }
        return x % MOD;
    }

    StaticModint() : x(0){}
    StaticModint(const my_type& a) : x(a.x){}
    StaticModint& operator=(const my_type&) = default;
    template< class Elem >
    StaticModint(Elem v) : x(safe_mod(v)){}
    unsigned int operator*() const { return x; }
    my_type& operator+=(const my_type& r) { auto t = x + r.x; if(t >= MOD) t -= MOD; x = t; return *this; }
    my_type operator+(const my_type& r) const { my_type res = *this; return res += r; }
    my_type& operator-=(const my_type& r) { auto t = x + MOD - r.x; if(t >= MOD) t -= MOD; x = t; return *this; }
    my_type operator-(const my_type& r) const { my_type res = *this; return res -= r; }
    my_type operator-() const noexcept { my_type res = *this; res.x = ((res.x == 0) ? 0 : (MOD - res.x)); return res; }
    my_type& operator*=(const my_type& r){ x = (u64)x * r.x % MOD; return *this; }
    my_type operator*(const my_type& r) const { my_type res = *this; return res *= r; }
    bool operator==(const my_type& r) const { return x == r.x; }
    my_type pow(unsigned long long i) const {
        my_type a = *this, res = 1;
        while(i){ if(i & 1){ res *= a; } a *= a; i >>= 1; }
        return res;
    }
    my_type inv() const { return my_type(ExtGcd(x, MOD).first); }
    unsigned int val() const { return x; }
    int hval() const { return int(x > MOD/2 ? x - MOD : x); }
    static constexpr unsigned int mod() { return MOD; }
    static my_type raw(unsigned int val) { auto res = my_type(); res.x = val; return res; }
    my_type& operator/=(const my_type& r){ return operator*=(r.inv()); }
    my_type operator/(const my_type& r) const { return operator*(r.inv()); }
};

} // namespace nachia
using Mint = nachia::StaticModint<998244353>;

namespace nachia{

template<unsigned int MOD>
struct PrimitiveRoot{
    using u64 = unsigned long long;
    static constexpr u64 powm(u64 a, u64 i) {
        u64 res = 1, aa = a;
        for( ; i; i /= 2){
            if(i & 1) res = res * aa % MOD;
            aa = aa * aa % MOD;
        }
        return res;
    }
    static constexpr bool ExamineVal(unsigned int g){
        u64 t = MOD - 1;
        for(u64 d=2; d*d<=t; d+=1+(d&1)) if(t % d == 0){
            if(powm(g, (MOD - 1) / d) == 1) return false;
            while(t % d == 0) t /= d;
        }
        if(t != 1) if(powm(g, (MOD - 1) / t) == 1) return false;
        return true;
    }
    static constexpr unsigned int GetVal(){
        for(u64 x=2; x<MOD; x++) if(ExamineVal(x)) return x;
        return 0;
    }
    static const unsigned int val = GetVal();
};

} // namespace nachia

namespace nachia{

int Popcount(unsigned long long c) noexcept {
#ifdef __GNUC__
    return __builtin_popcountll(c);
#else
    c = (c & (~0ull/3)) + ((c >> 1) & (~0ull/3));
    c = (c & (~0ull/5)) + ((c >> 2) & (~0ull/5));
    c = (c & (~0ull/17)) + ((c >> 4) & (~0ull/17));
    c = (c * (~0ull/257)) >> 56;
    return c;
#endif
}

// please ensure x != 0
int MsbIndex(unsigned long long x) noexcept {
#ifdef __GNUC__
    return 63 - __builtin_clzll(x);
#else
    using u64 = unsigned long long;
    int q = (x >> 32) ? 32 : 0;
    auto m = x >> q;
    constexpr u64 hi = 0x88888888;
    constexpr u64 mi = 0x11111111;
    m = (((m | ~(hi - (m & ~hi))) & hi) * mi) >> 35;
    m = (((m | ~(hi - (m & ~hi))) & hi) * mi) >> 31;
    q += (m & 0xf) << 2;
    q += 0x3333333322221100 >> (((x >> q) & 0xf) << 2) & 0xf;
    return q;
#endif
}

// please ensure x != 0
int LsbIndex(unsigned long long x) noexcept {
#ifdef __GNUC__
    return __builtin_ctzll(x);
#else
    return MsbIndex(x & -x);
#endif
}

}


namespace nachia {

template<class mint>
struct NttInterface{

template<class Iter>
void Butterfly(Iter, int) const {}

template<class Iter>
void IButterfly(Iter, int) const {}

template<class Iter>
void BitReversal(Iter a, int N) const {
    for(int i=0, j=0; j<N; j++){
        if(i < j) std::swap(a[i], a[j]);
        for(int k = N>>1; k > (i^=k); k>>=1);
    }
}

};

} // namespace nachia
#include <iterator>
#include <array>

namespace nachia{

template <class mint>
struct Ntt : NttInterface<mint> {

using u32 = unsigned int;
using u64 = unsigned long long;
    
static int ceil_pow2(int n) {
    int x = 0;
    while ((1U << x) < (u32)(n)) x++;
    return x;
}
    
static constexpr int bsf_constexpr(unsigned int n) {
    int x = 0;
    while (!(n & (1 << x))) x++;
    return x;
}

struct fft_info {
    static constexpr u32 g = nachia::PrimitiveRoot<mint::mod()>::val;
    static constexpr int rank2 = bsf_constexpr(mint::mod()-1);
    using RootTable = std::array<mint, rank2+1>;
    RootTable root, iroot, rate3, 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<=rank2-3; i++){
            rate3[i] = root[i+3] * prod;
            irate3[i] = iroot[i+3] * iprod;
            prod *= iroot[i+3];
            iprod *= root[i+3];
        }
    }
};

template<class RandomAccessIterator>
void ButterflyLayered(RandomAccessIterator a, int n, int stride, int repeat) const {
    static const fft_info info;
    int h = n * stride;
    
    while(repeat--){

    int len = 1;
    int p = h;
    if(ceil_pow2(n)%2 == 1){
        p >>= 1;
        for(int i=0; i<p; i++){
            mint l = a[i], r = a[i+p];
            a[i] = l+r; a[i+p] = l-r;
        }
        len <<= 1;
    }
    for( ; p > stride; ){
        p >>= 2;
        mint rot = 1, imag = info.root[2];
        u64 mod2 = u64(mint::mod()) * mint::mod();
        int offset = p;
        for(int s=0; s<len; s++){
            if(s) rot *= info.rate3[LsbIndex(~(u32)(s-1))];
            mint rot2 = rot * rot;
            mint rot3 = rot2 * rot;
            for(int i=offset-p; i<offset; i++){
                u64 a0 = u64(a[i].val());
                u64 a1 = u64(a[i+p].val()) * rot.val();
                u64 a2 = u64(a[i+2*p].val()) * rot2.val();
                u64 a3 = u64(a[i+3*p].val()) * rot3.val();
                u64 a1na3imag = u64(mint(a1 + mod2 - a3).val()) * imag.val();
                u64 na2 = mod2 - a2;
                a[i] = a0 + a2 + a1 + a3;
                a[i+1*p] = a0 + a2 + (2 * mod2 - (a1 + a3));
                a[i+2*p] = a0 + na2 + a1na3imag;
                a[i+3*p] = a0 + na2 + (mod2 - a1na3imag);
            }
            offset += p << 2;
        }
        len <<= 2;
    }
    
    a += h;
    }
}

template<class RandomAccessIterator>
void Butterfly(RandomAccessIterator a, int n) const {
    ButterflyLayered(a, n, 1, 1);
}

template<class RandomAccessIterator>
void IButterflyLayered(RandomAccessIterator a, int n, int stride, int repeat) const {

    static const fft_info info;
    constexpr int MOD = mint::mod();
    
    while(repeat--){
    
    int len = n;
    int p = stride;

    for( ; 2 < len; ){
        len >>= 2;
        mint irot = 1, iimag = info.iroot[2];
        int offset = p;
        for(int s=0; s<len; s++){
            if(s) irot *= info.irate3[LsbIndex(~(u32)(s-1))];
            mint irot2 = irot * irot;
            mint irot3 = irot2 * irot;
            for(int i=offset-p; i<offset; i++){
                u64 a0 = a[i].val();
                u64 a1 = a[i+p].val();
                u64 a2 = a[i+2*p].val();
                u64 a3 = a[i+3*p].val();
                u64 a2na3iimag = mint((a2 + MOD - a3) * iimag.val()).val();
                a[i] = a0 + a1 + a2 + a3;
                a[i+p] = (a0 + (MOD - a1) + a2na3iimag) * irot.val();
                a[i+2*p] = (a0 + a1 + (MOD - a2) + (MOD - a3)) * irot2.val();
                a[i+3*p] = (a0 + (MOD - a1) + (MOD - a2na3iimag)) * irot3.val();
            }
            offset += p << 2;
        }
        p <<= 2;
    }
    if(len == 2){
        for(int i=0; i<p; i++){
            mint l = a[i], r = a[i+p];
            a[i] = l+r; a[i+p] = l-r;
        }
        p <<= 1;
    }
    
    a += p;
    }
}

template<class RandomAccessIterator>
void IButterfly(RandomAccessIterator a, int n) const {
    IButterflyLayered(a, n, 1, 1);
}

};

} // namespace nachia

namespace nachia{

template<class Modint>
std::vector<Modint> Convolution(
    const std::vector<Modint>& A,
    const std::vector<Modint>& B
){
    int n = A.size();
    int m = B.size();
    if(n <= 40 || m <= 40){
        if(n+m == 0) return std::vector<Modint>(0);
        std::vector<Modint> C(n+m-1);
        for(int i=0; i<n; i++) for(int j=0; j<m; j++){
            C[i+j] += A[i] * B[j];
        }
        return C;
    }
    int g = 1; while(g < n+m-1) g *= 2;
    std::vector<Modint> C(g);
    for(int i=0; i<n; i++) C[i] = A[i];
    Ntt<Modint> ntt;
    ntt.Butterfly(C.begin(), g);
    std::vector<Modint> D(g);
    for(int i=0; i<m; i++) D[i] = B[i];
    ntt.Butterfly(D.begin(), g);
    Modint w = Modint(g).inv();
    for(int i=0; i<g; i++) C[i] *= D[i] * w;
    ntt.IButterfly(C.begin(), g);
    C.resize(n+m-1);
    return C;
}

} // namespace nachia

// N = A.size();
// for 0 <= k < M {
//    res[k] := sum_{0 <= i < N} A[i] * X[i]**k;
// }
// return res;

namespace nachia{

template<class Modint>
std::vector<Modint> WeightedPowerSum(std::vector<Modint> A, std::vector<Modint> X, int M){
    auto invFps = [](std::vector<Modint> B, int n){
        Modint b = B[0].inv();
        for(auto& a : B) a *= b;
        std::vector<Modint> res = {1};
        int z = 1;
        while(z < n){
            std::vector<Modint> T(z*2);
            for(int i=1; i<z*2 && i<(int)B.size(); i++) T[i] = -B[i];
            T = Convolution(res, T);
            T = std::vector(T.begin() + z, T.begin() + z*2);
            T = Convolution(T, res);
            for(int i=0; i<z; i++) res.push_back(T[i]);
            z *= 2;
        }
        res.resize(n);
        for(auto& a : res) a *= b;
        return res;
    };
    std::vector<std::pair<std::vector<Modint>, std::vector<Modint>>> fracs(A.size()*2);
    for(std::size_t i=0; i<A.size(); i++){
        fracs[i].first = std::vector<Modint>{ A[i] }; 
        fracs[i].second = std::vector<Modint>{ 1, -X[i] };
    }
    std::size_t l = 0;
    std::size_t r = A.size();
    while(l + 1 != r){
        fracs[r].second = Convolution(fracs[l].second, fracs[l+1].second);
        auto a = Convolution(fracs[l].first, fracs[l+1].second);
        auto b = Convolution(fracs[l].second, fracs[l+1].first);
        if(a.size() < b.size()) std::swap(a,b);
        for(std::size_t i=0; i<b.size(); i++) a[i] += b[i];
        std::swap(fracs[r].first, a);
        l += 2; r++;
    }
    auto res = Convolution(fracs[l].first, invFps(fracs[l].second, M));
    res.resize(M);
    return res;
}

template<class Modint>
std::vector<Modint> WeightedExpAxSum(std::vector<Modint> W, std::vector<Modint> A, int M){
    if(M == 0) return std::vector<Modint>();
    auto buf = WeightedPowerSum(std::move(W), std::move(A), M);
    std::vector<Modint> fact(M);
    fact[0] = Modint(1);
    for(int i=1; i<M; i++) fact[i] = fact[i-1] * Modint(i);
    Modint f = fact[M-1].inv();
    for(int i=M-1; i>=0; i--){ buf[i] *= f; f += Modint(i); }
    return fact;
}

// x <- e^x
template<class Modint>
std::vector<Modint> SubstituteExpX(std::vector<Modint> F, int M){
    std::vector<Modint> x(M);
    for(int i=0; i<M; i++) x[i] = Modint(i);
    return WeightedExpAxSum(std::move(F), std::move(x), M);
}

} // namespace nachia

void testcase(){
  ll N, M; cin >> N >> M;
  V<Mint> A(N); REP(i,N){ ll a; cin >> a; A[i] = a; }
  auto B = nachia::WeightedPowerSum(V<Mint>(N,1), A, M+1);
  V<Mint> mobius(M+1); mobius[1] = 1;
  for(ll p=1; p<=M; p++) for(ll q=2; p*q<=M; q++) mobius[p*q] -= mobius[p];
  V<Mint> X(M+1);
  for(ll p=1; p<=M; p++) for(ll q=1; p*q<=M; q++){
    X[p*q] += mobius[p] * B[p].pow(q);
  }
  REP(i,M){
    if(i) cout << " ";
    cout << X[i+1].val();
  } cout << "\n";
}

int main(){
  cin.tie(0)->sync_with_stdio(0);
  testcase();
  return 0;
}