FormalPowerSeries
(src/fps/formal_power_series.hpp)

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• Last update: 2024-11-09 02:16:49+09:00
• Include: #include "src/fps/formal_power_series.hpp"

FormalPowerSeries

形式的冪級数の計算を行うライブラリです．

$\mathrm{mod}$ が $\mathrm{NTT-friendly}$ なときのみ使えます．
そうでない $\mathrm{mod}$ や long long の範囲で形式的冪級数を扱いたい場合は FormalPowerSeriesArbitrary や FormalPowerSeriesLL を使ってください．

基本的な使い方としては以下のとおりです．

#include "template/template.hpp"
#include "template/static_modint.hpp"
#include "fps/formal_power_series.hpp"
using mint = modint998244353;
using fps = FormalPowerSeries<mint>;
int main(void) {
    fps f(10), g(10);
    for(int i = 0; i < 10; i++) {
        f[i] = i;
        g[i] = i * 2;
    }
    fps h = f * g;
    for(int i = 0; i < 19; i++) {
        cout << h[i] << '\n';
    }
}

コンストラクタ

fps f(int n)

$n - 1$ 次の多項式 $f$ を作ります．
初期値は $f(x) = 0 + 0x + 0x^2 + … + 0x^{n - 1}$ です．

制約

• $0 \leq n$

計算量

• $O(n)$

eval

mint f.eval(mint a)

$f(a)$ を返します．

計算量

• $O(n)$

shrink

void f.shrink()

係数が $0$ となっている高次の項を削除します．

たとえば $f(x) = 1 + 0x + 3x^2 + 0x^3 + 0x^4$ なら $f(x) = 1 + 0x + 3x^2$ となります．

計算量

• $O(n)$

pre

fps f.pre(int deg)

$f(x)$ の deg 次以下の項を取り出します．
$\mathrm{deg} > n$ の場合は多項式が拡張されます．

制約

• $0 \leq \mathrm{deg}$

計算量

• $O(\mathrm{deg})$

rev

fps f.rev()

係数を逆順にします．

たとえば $f(x) = 1 + 2x + 3x^2$ なら $f(x) = 3 + 2x + 1x^2$ となります．

計算量

• $O(n)$

各種演算

fps = fps;

-fps;

fps += mint;
fps -= mint;
fps *= mint;
fps /= mint;

fps += fps;
fps -= fps;
fps *= fps;
fps /= fps;
fps %= fps;

fps + mint;
fps - mint;
fps * mint;
fps / mint;

fps + fps;
fps - fps;
fps * fps;
fps / fps;
fps % fps;

fps << int;
fps >> int;

が動きます．

サイズの調整などは適宜行なってくれます．

fps / fps および fps % fps は，多項式としての除算です．
すなわち $f(x) = g(x) q(x) + r(x)$ と表せるとき，
f(x) / g(x) = q(x) ， f(x) % g(x) = r(x) です．

また fps << (int d) は $f(x) * x^d$ です．
fps >> (int d) は $f(x) / x^d$ です．

制約

• 除算・剰余演算で未定義な演算をしないこと

計算量

• $O(1)$ (多項式とmintの加算・減算)
• $O(n \log n)$ (多項式同士の乗算・除算・剰余演算)
• $O(n)$ (その他)

onemul

void f.onemul(int d, mint c, int deg = -1)

$f(x) = (1 + cx^d) f(x)$ とします．

deg で何次まで計算するかを指定できます．
deg = -1 のときは deg = n + d とします．
$\mathrm{deg} > n + d$ のときは多項式が拡張されます．

制約

• $-1 \leq deg$

計算量

• $O(n + d)$ ( deg = -1 のとき)

onediv

void f.onediv(int d, mint c, int deg = -1)

$f(x) = \frac{f(x)}{1 + cx^d}$ とします．

deg で何次まで計算するかを指定できます．
deg = -1 のときは deg = n とします．
$\mathrm{deg} > n + d$ のときは多項式が拡張されます．

制約

• $-1 \leq \mathrm{deg}$

計算量

• $O(n)$ ( deg = -1 のとき)

diff

fps f.diff()

$f’(x)$ を返します．

計算量

• $O(n)$

integral

fps f.integral()

$\int f(x) dx$ を返します．
積分定数は $0$ としてあります．

計算量

• $O(n)$

inv

fps f.inv(int deg = -1)

$f(x) g(x) = 1 \pmod{x^n}$ を満たす $g(x)$ を返します．

deg で何次まで計算するかを指定できます．
deg = -1 のときは deg = n とします．

制約

• $0 \leq n$
• $f(x)$ の定数項は $0$ でない．

計算量

• $O(n \log n)$ ( deg = -1 のとき)

log

fps f.log(int deg = -1)

$\log f(x)$ を返します．
$\log f(x)$ とは

\[f(x) = e^{g(x)} ~ ( := \sum\limits_{k=0}^{n-1} \frac{g^k(x)}{k!} \pmod{x^n})\]

を満たす $g(x) = \sum\limits_{i=0}^{n-1} b_i x^i (b_0 = 0)$ のことです．

deg で何次まで計算するかを指定できます．
deg = -1 のときは deg = n とします．

制約

• $0 < n$
• $f(x)$ の定数項は $1$ である．

計算量

• $O(n \log n)$ ( deg = -1 のとき)

exp

fps f.exp(int deg = -1)

$e^{f(x)} := \sum\limits_{i=0}^{n-1} \frac{f^i(x)}{i!} \pmod{x^n}$ を返します．

deg で何次まで計算するかを指定できます．
deg = -1 のときは deg = n とします．

制約

• $n = 0$ または $f(x)$ の定数項が $0$ である．

計算量

• $O(n \log n)$ ( deg = -1 のとき)

pow

fps f.pow(ll k, int deg = -1)

$f^k(x) \pmod{x^n}$ を返します．

deg で何次まで計算するかを指定できます．
deg = -1 のときは deg = n とします．

制約

• $0 \leq k$
• $-1 \leq \mathrm{deg}$

計算量

• $O(n \log n)$ ( deg = -1 のとき)

shift

fps f.shift(mint c)

$f(x + a)$ を返します．

計算量

• $O(n \log n)$

Depends on

• convolution (src/convolution/convolution.hpp)
• pow_mod (src/math/pow_mod.hpp)
• primitive_root (src/math/primitive_root.hpp)
• template (src/template/template.hpp)

Verified with

• verify/library_checker/other/find_linear_recurrence.test.cpp
• verify/library_checker/other/kth_term_of_linearly_recurrent_sequence.test.cpp
• verify/library_checker/polynomial/division_of_polynomial.test.cpp
• verify/library_checker/polynomial/exp_of_formal_power_series.test.cpp
• verify/library_checker/polynomial/inv_of_formal_power_series.test.cpp
• verify/library_checker/polynomial/log_of_formal_power_series.test.cpp
• verify/library_checker/polynomial/polynomial_taylor_shift.test.cpp
• verify/library_checker/polynomial/pow_of_formal_power_series.test.cpp
• verify/library_checker/polynomial/product_of_polynomial_sequence.test.cpp

Code

#pragma once
#include "../template/template.hpp"
#include "../convolution/convolution.hpp"
template <typename mint>
struct FormalPowerSeries : vector<mint> {
    using vector<mint>::vector;
    using F = FormalPowerSeries;
    F& operator=(const vector<mint>& g) {
        const int n = (*this).size();
        const int m = g.size();
        if(n < m) (*this).resize(m);
        for(int i = 0; i < m; ++i) (*this)[i] = g[i];
        return (*this);
    }
    F& operator-() {
        const int n = (*this).size();
        for(int i = 0; i < n; ++i) (*this)[i] *= -1;
        return (*this);
    }
    F& operator+=(const F& g) {
        const int n = (*this).size();
        const int m = g.size();
        if(n < m) (*this).resize(m);
        for(int i = 0; i < m; ++i) (*this)[i] += g[i];
        return (*this);
    }
    F& operator+=(const mint& r) {
        if((*this).empty()) (*this).resize(1, mint(0));
        (*this)[0] += r;
        return (*this);
    }
    F& operator-=(const F& g) {
        const int n = (*this).size();
        const int m = g.size();
        if(n < m) (*this).resize(m);
        for(int i = 0; i < m; ++i) (*this)[i] -= g[i];
        return (*this);
    }
    F& operator-=(const mint& r) {
        if((*this).empty()) (*this).resize(1, mint(0));
        (*this)[0] -= r;
        return (*this);
    }
    F& operator*=(const F& g) {
        (*this) = convolution((*this), g);
        return (*this);
    }
    F& operator*=(const mint& r) {
        const int n = (*this).size();
        for(int i = 0; i < n; ++i) (*this)[i] *= r;
        return (*this);
    }
    F& operator/=(const F& g) {
        if((*this).size() < g.size()) {
            (*this).clear();
            return (*this);
        }
        const int n = (*this).size() - g.size() + 1;
        (*this) = ((*this).rev().pre(n) * g.rev().inv(n)).pre(n).rev();
        return (*this);
    }
    F& operator/=(const mint& r) {
        const int n = (*this).size();
        const mint inv_r = r.inv();
        for(int i = 0; i < n; ++i) (*this)[i] *= inv_r;
        return (*this);
    }
    F& operator%=(const F& g) {
        (*this) -= (*this) / g * g;
        shrink();
        return (*this);
    }
    F operator*(const mint& g) const {
        return F(*this) *= g;
    }
    F operator-(const mint& g) const {
        return F(*this) -= g;
    }
    F operator+(const mint& g) const {
        return F(*this) += g;
    }
    F operator/(const mint& g) const {
        return F(*this) /= g;
    }
    F operator*(const F& g) const {
        return F(*this) *= g;
    }
    F operator-(const F& g) const {
        return F(*this) -= g;
    }
    F operator+(const F& g) const {
        return F(*this) += g;
    }
    F operator/(const F& g) const {
        return F(*this) /= g;
    }
    F operator%(const F& g) const {
        return F(*this) %= g;
    }
    F operator<<(const int d) const {
        F ret(*this);
        ret.insert(ret.begin(), d, mint(0));
        return ret;
    }
    F operator>>(const int d) const {
        const int n = (*this).size();
        if(n <= d) return {};
        F ret(*this);
        ret.erase(ret.begin(), ret.begin() + d);
        return ret;
    }
    void shrink() {
        while(!(*this).empty() and (*this).back() == mint(0)) (*this).pop_back();
    }
    F rev() const {
        F ret(*this);
        reverse(begin(ret), end(ret));
        return ret;
    }
    F pre(const int deg) const {
        assert(deg >= 0);
        F ret(begin(*this), begin(*this) + min((int)(*this).size(), deg));
        if((int)ret.size() < deg) ret.resize(deg);
        return ret;
    }
    mint eval(const mint& a) const {
        const int n = (*this).size();
        mint x = 1, ret = 0;
        for(int i = 0; i < n; ++i) {
            ret += (*this)[i] * x;
            x *= a;
        }
        return ret;
    }
    void onemul(const int d, const mint& c, int deg = -1) {
        assert(deg >= -1);
        const int n = (*this).size();
        if(deg == -1) deg = n + d;
        if(deg > n) (*this).resize(deg);
        for(int i = deg - d - 1; i >= 0; --i) {
            (*this)[i + d] += (*this)[i] * c;
        }
    }
    void onediv(const int d, const mint& c, int deg = -1) {
        assert(deg >= -1);
        const int n = (*this).size();
        if(deg == -1) deg = n;
        if(deg > n) (*this).resize(deg + 1);
        for(int i = 0; i < deg - d; ++i) {
            (*this)[i + d] -= (*this)[i] * c;
        }
    }
    F diff() const {
        const int n = (*this).size();
        F ret(max(0, n - 1));
        for(int i = 1; i < n; ++i) ret[i - 1] = (*this)[i] * i;
        return ret;
    }
    F integral() const {
        const int n = (*this).size();
        static constexpr int mod = mint::mod();
        F ret(n + 1);
        ret[0] = mint(0);
        if(n > 0) ret[1] = mint(1);
        for(int i = 2; i <= n; ++i) ret[i] = (-ret[mod % i]) * (mod / i);
        for(int i = 0; i < n; ++i) ret[i + 1] *= (*this)[i];
        return ret;
    }
    F inv(int deg = -1) const {
        assert(deg >= -1);
        const int n = (*this).size();
        assert(n > 0 and (*this)[0] != mint(0));
        if(deg == -1) deg = n;
        F g(1);
        g[0] = (*this)[0].inv();
        while((int)g.size() < deg) {
            const int m = g.size();
            F f(begin(*this), begin(*this) + min(n, 2 * m));
            F r(g);
            f.resize(2 * m);
            r.resize(2 * m);
            butterfly(f);
            butterfly(r);
            for(int i = 0; i < 2 * m; ++i) f[i] *= r[i];
            butterfly_inv(f);
            f.erase(f.begin(), f.begin() + m);
            f.resize(2 * m);
            butterfly(f);
            for(int i = 0; i < 2 * m; ++i) f[i] *= r[i];
            butterfly_inv(f);
            mint in = mint(2 * m).inv();
            in *= -in;
            for(int i = 0; i < m; ++i) f[i] *= in;
            g.insert(g.end(), f.begin(), f.begin() + m);
        }
        return g.pre(deg);
    }
    F log(int deg = -1) const {
        assert(deg >= -1);
        const int n = (*this).size();
        assert(n > 0 and (*this)[0] == mint(1));
        if(deg == -1) deg = n;
        return ((*this).diff() * (*this).inv(deg)).pre(deg - 1).integral();
    }
    F exp(int deg = -1) const {
        assert(deg >= -1);
        const int n = (*this).size();
        assert(n == 0 or (*this)[0] == 0);
        if(deg == -1) deg = n;
        F Inv;
        Inv.reserve(deg + 1);
        Inv.emplace_back(mint(0));
        Inv.emplace_back(mint(1));
        auto inplace_integral = [&](F& f) -> void {
            const int n = (int)f.size();
            static constexpr int mod = mint::mod();
            while((int)Inv.size() <= n) {
                const int i = Inv.size();
                Inv.emplace_back((-Inv[mod % i]) * (mod / i));
            }
            f.insert(begin(f), mint(0));
            for(int i = 1; i <= n; ++i) f[i] *= Inv[i];
        };
        auto inplace_diff = [](F& f) -> void {
            if(f.empty()) return;
            f.erase(begin(f));
            mint coeff = 1;
            for(int i = 0; i < (int)f.size(); ++i) {
                f[i] *= coeff;
                ++coeff;
            }
        };
        F b{1, 1 < (int)(*this).size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
        for(int m = 2; m < deg; m <<= 1) {
            auto y = b;
            y.resize(2 * m);
            butterfly(y);
            z1 = z2;
            F z(m);
            for(int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
            butterfly_inv(z);
            const mint si = mint(m).inv();
            for(int i = 0; i < m; ++i) z[i] *= si;
            fill(begin(z), begin(z) + m / 2, mint(0));
            butterfly(z);
            for(int i = 0; i < m; ++i) z[i] *= -z1[i];
            butterfly_inv(z);
            for(int i = 0; i < m; ++i) z[i] *= si;
            c.insert(end(c), begin(z) + m / 2, end(z));
            z2 = c;
            z2.resize(2 * m);
            butterfly(z2);
            F x(begin((*this)), begin((*this)) + min<int>((*this).size(), m));
            x.resize(m);
            inplace_diff(x);
            x.emplace_back(mint(0));
            butterfly(x);
            for(int i = 0; i < m; ++i) x[i] *= y[i];
            butterfly_inv(x);
            for(int i = 0; i < m; ++i) x[i] *= si;
            x -= b.diff();
            x.resize(2 * m);
            for(int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0);
            butterfly(x);
            for(int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
            butterfly_inv(x);
            const mint si2 = mint(m << 1).inv();
            for(int i = 0; i < 2 * m; ++i) x[i] *= si2;
            x.pop_back();
            inplace_integral(x);
            for(int i = m; i < min<int>((*this).size(), 2 * m); ++i) x[i] += (*this)[i];
            fill(begin(x), begin(x) + m, mint(0));
            butterfly(x);
            for(int i = 0; i < 2 * m; ++i) x[i] *= y[i];
            butterfly_inv(x);
            for(int i = 0; i < 2 * m; ++i) x[i] *= si2;
            b.insert(end(b), begin(x) + m, end(x));
        }
        return b.pre(deg);
    }
    F pow(const long long k, int deg = -1) const {
        assert(deg >= -1);
        assert(k >= 0);
        const int n = (*this).size();
        if(deg == -1) deg = n;
        if(k == 0) {
            F ret(deg);
            if(deg) ret[0] = 1;
            return ret;
        }
        for(int i = 0; i < n; ++i) {
            if((*this)[i] != mint(0)) {
                const mint rev = mint(1) / (*this)[i];
                F ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
                ret *= (*this)[i].pow(k);
                ret = (ret << (i * k)).pre(deg);
                if((int)ret.size() < deg) ret.resize(deg, mint(0));
                return ret;
            }
            if(__int128_t(i + 1) * k >= deg) return F(deg, mint(0));
        }
        return F(deg, mint(0));
    }
    F shift(const mint& c) const {
        const int n = (*this).size();
        vector<mint> fact(n), ifact(n);
        fact[0] = ifact[0] = mint(1);
        for(int i = 1; i < n; ++i) fact[i] = fact[i - 1] * i;
        ifact[n - 1] = mint(1) / fact[n - 1];
        for(int i = n - 1; i > 1; --i) ifact[i - 1] = ifact[i] * i;
        F ret(*this);
        for(int i = 0; i < n; ++i) ret[i] *= fact[i];
        ret = ret.rev();
        F bs(n, mint(1));
        for(int i = 1; i < n; ++i) bs[i] = bs[i - 1] * c * ifact[i] * fact[i - 1];
        ret = (ret * bs).pre(n);
        ret = ret.rev();
        for(int i = 0; i < n; ++i) ret[i] *= ifact[i];
        return ret;
    }
};

#line 2 "src/template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
using ll = long long;
using P = pair<long long, long long>;
#define rep(i, a, b) for(long long i = (a); i < (b); ++i)
#define rrep(i, a, b) for(long long i = (a); i >= (b); --i)
constexpr long long inf = 4e18;
struct SetupIO {
    SetupIO() {
        ios::sync_with_stdio(0);
        cin.tie(0);
        cout << fixed << setprecision(30);
    }
} setup_io;
#line 3 "src/math/pow_mod.hpp"
constexpr long long pow_mod(long long x, long long n, const long long mod) {
    assert(n >= 0 and mod >= 1);
    x %= mod;
    if(x < 0) x += mod;
    long long res = 1;
    while(n > 0) {
        if(n & 1) res = res * x % mod;
        x = x * x % mod;
        n >>= 1;
    }
    return res;
}
#line 4 "src/math/primitive_root.hpp"
constexpr int primitive_root(const 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(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if(ok) return g;
    }
}
#line 4 "src/convolution/convolution.hpp"
constexpr int countr_zero(const unsigned int n) {
    int res = 0;
    while(!(n & (1 << res))) ++res;
    return res;
}
template <typename mint, int g = primitive_root(mint::mod())>
struct FFT_Info {
    static constexpr int rank2 = countr_zero(mint::mod() - 1);
    array<mint, rank2 + 1> root;
    array<mint, rank2 + 1> iroot;
    array<mint, max(0, rank2 - 2 + 1)> rate2;
    array<mint, max(0, rank2 - 2 + 1)> irate2;
    array<mint, max(0, rank2 - 3 + 1)> rate3;
    array<mint, max(0, rank2 - 3 + 1)> 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 - 2; ++i) {
                rate2[i] = root[i + 2] * prod;
                irate2[i] = iroot[i + 2] * iprod;
                prod *= iroot[i + 2];
                iprod *= root[i + 2];
            }
        }
        {
            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 <typename mint>
void butterfly(vector<mint>& a) {
    const int n = (int)a.size();
    const int h = __builtin_ctz((unsigned int)n);
    static const FFT_Info<mint> info;
    int len = 0;
    while(len < h) {
        if(h - len == 1) {
            const int p = 1 << (h - len - 1);
            mint rot = 1;
            for(int s = 0; s < (1 << len); ++s) {
                const int offset = s << (h - len);
                for(int i = 0; i < p; ++i) {
                    const auto l = a[i + offset];
                    const auto r = a[i + offset + p] * rot;
                    a[i + offset] = l + r;
                    a[i + offset + p] = l - r;
                }
                if(s + 1 != (1 << len)) rot *= info.rate2[__builtin_ctz(~(unsigned int)(s))];
            }
            ++len;
        } else {
            const int p = 1 << (h - len - 2);
            mint rot = 1, imag = info.root[2];
            for(int s = 0; s < (1 << len); ++s) {
                const mint rot2 = rot * rot;
                const mint rot3 = rot2 * rot;
                const int offset = s << (h - len);
                for(int i = 0; i < p; ++i) {
                    const auto mod2 = 1ULL * mint::mod() * mint::mod();
                    const auto a0 = 1ULL * a[i + offset].val();
                    const auto a1 = 1ULL * a[i + offset + p].val() * rot.val();
                    const auto a2 = 1ULL * a[i + offset + 2 * p].val() * rot2.val();
                    const auto a3 = 1ULL * a[i + offset + 3 * p].val() * rot3.val();
                    const auto a1na3imag = 1ULL * mint(a1 + mod2 - a3).val() * imag.val();
                    const auto na2 = mod2 - a2;
                    a[i + offset] = a0 + a2 + a1 + a3;
                    a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
                    a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
                    a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
                }
                if(s + 1 != (1 << len)) rot *= info.rate3[__builtin_ctz(~(unsigned int)(s))];
            }
            len += 2;
        }
    }
}
template <typename mint>
void butterfly_inv(vector<mint>& a) {
    const int n = (int)a.size();
    const int h = __builtin_ctz((unsigned int)n);
    static const FFT_Info<mint> info;
    int len = h;
    while(len) {
        if(len == 1) {
            const int p = 1 << (h - len);
            mint irot = 1;
            for(int s = 0; s < (1 << (len - 1)); ++s) {
                const int offset = s << (h - len + 1);
                for(int i = 0; i < p; ++i) {
                    const auto l = a[i + offset];
                    const auto r = a[i + offset + p];
                    a[i + offset] = l + r;
                    a[i + offset + p] = (unsigned long long)(mint::mod() + l.val() - r.val()) * irot.val();
                }
                if(s + 1 != (1 << (len - 1))) irot *= info.irate2[__builtin_ctz(~(unsigned int)(s))];
            }
            --len;
        } else {
            const int p = 1 << (h - len);
            mint irot = 1, iimag = info.iroot[2];
            for(int s = 0; s < (1 << (len - 2)); ++s) {
                const mint irot2 = irot * irot;
                const mint irot3 = irot2 * irot;
                const int offset = s << (h - len + 2);
                for(int i = 0; i < p; ++i) {
                    const auto a0 = 1ULL * a[i + offset + 0 * p].val();
                    const auto a1 = 1ULL * a[i + offset + 1 * p].val();
                    const auto a2 = 1ULL * a[i + offset + 2 * p].val();
                    const auto a3 = 1ULL * a[i + offset + 3 * p].val();
                    const auto a2na3iimag = 1ULL * mint((mint::mod() + a2 - a3) * iimag.val()).val();
                    a[i + offset] = a0 + a1 + a2 + a3;
                    a[i + offset + 1 * p] = (a0 + (mint::mod() - a1) + a2na3iimag) * irot.val();
                    a[i + offset + 2 * p] = (a0 + a1 + (mint::mod() - a2) + (mint::mod() - a3)) * irot2.val();
                    a[i + offset + 3 * p] = (a0 + (mint::mod() - a1) + (mint::mod() - a2na3iimag)) * irot3.val();
                }
                if(s + 1 != (1 << (len - 2))) irot *= info.irate3[__builtin_ctz(~(unsigned int)(s))];
            }
            len -= 2;
        }
    }
}
template <typename mint>
vector<mint> convolution_naive(const vector<mint>& a, const vector<mint>& b) {
    const int n = (int)a.size(), m = (int)b.size();
    vector<mint> res(n + m - 1);
    if(n < m) {
        for(int j = 0; j < m; ++j) {
            for(int i = 0; i < n; ++i) {
                res[i + j] += a[i] * b[j];
            }
        }
    } else {
        for(int i = 0; i < n; ++i) {
            for(int j = 0; j < m; ++j) {
                res[i + j] += a[i] * b[j];
            }
        }
    }
    return res;
}
template <typename mint>
vector<mint> convolution(vector<mint> a, vector<mint> b) {
    const int n = (int)a.size(), m = (int)b.size();
    if(n == 0 or m == 0) return {};
    int z = 1;
    while(z < n + m - 1) z *= 2;
    assert((mint::mod() - 1) % z == 0);
    if(min(n, m) <= 60) return convolution_naive(a, b);
    a.resize(z);
    b.resize(z);
    butterfly(a);
    butterfly(b);
    for(int i = 0; i < z; ++i) a[i] *= b[i];
    butterfly_inv(a);
    a.resize(n + m - 1);
    const mint iz = mint(z).inv();
    for(int i = 0; i < n + m - 1; ++i) a[i] *= iz;
    return a;
}
#line 4 "src/fps/formal_power_series.hpp"
template <typename mint>
struct FormalPowerSeries : vector<mint> {
    using vector<mint>::vector;
    using F = FormalPowerSeries;
    F& operator=(const vector<mint>& g) {
        const int n = (*this).size();
        const int m = g.size();
        if(n < m) (*this).resize(m);
        for(int i = 0; i < m; ++i) (*this)[i] = g[i];
        return (*this);
    }
    F& operator-() {
        const int n = (*this).size();
        for(int i = 0; i < n; ++i) (*this)[i] *= -1;
        return (*this);
    }
    F& operator+=(const F& g) {
        const int n = (*this).size();
        const int m = g.size();
        if(n < m) (*this).resize(m);
        for(int i = 0; i < m; ++i) (*this)[i] += g[i];
        return (*this);
    }
    F& operator+=(const mint& r) {
        if((*this).empty()) (*this).resize(1, mint(0));
        (*this)[0] += r;
        return (*this);
    }
    F& operator-=(const F& g) {
        const int n = (*this).size();
        const int m = g.size();
        if(n < m) (*this).resize(m);
        for(int i = 0; i < m; ++i) (*this)[i] -= g[i];
        return (*this);
    }
    F& operator-=(const mint& r) {
        if((*this).empty()) (*this).resize(1, mint(0));
        (*this)[0] -= r;
        return (*this);
    }
    F& operator*=(const F& g) {
        (*this) = convolution((*this), g);
        return (*this);
    }
    F& operator*=(const mint& r) {
        const int n = (*this).size();
        for(int i = 0; i < n; ++i) (*this)[i] *= r;
        return (*this);
    }
    F& operator/=(const F& g) {
        if((*this).size() < g.size()) {
            (*this).clear();
            return (*this);
        }
        const int n = (*this).size() - g.size() + 1;
        (*this) = ((*this).rev().pre(n) * g.rev().inv(n)).pre(n).rev();
        return (*this);
    }
    F& operator/=(const mint& r) {
        const int n = (*this).size();
        const mint inv_r = r.inv();
        for(int i = 0; i < n; ++i) (*this)[i] *= inv_r;
        return (*this);
    }
    F& operator%=(const F& g) {
        (*this) -= (*this) / g * g;
        shrink();
        return (*this);
    }
    F operator*(const mint& g) const {
        return F(*this) *= g;
    }
    F operator-(const mint& g) const {
        return F(*this) -= g;
    }
    F operator+(const mint& g) const {
        return F(*this) += g;
    }
    F operator/(const mint& g) const {
        return F(*this) /= g;
    }
    F operator*(const F& g) const {
        return F(*this) *= g;
    }
    F operator-(const F& g) const {
        return F(*this) -= g;
    }
    F operator+(const F& g) const {
        return F(*this) += g;
    }
    F operator/(const F& g) const {
        return F(*this) /= g;
    }
    F operator%(const F& g) const {
        return F(*this) %= g;
    }
    F operator<<(const int d) const {
        F ret(*this);
        ret.insert(ret.begin(), d, mint(0));
        return ret;
    }
    F operator>>(const int d) const {
        const int n = (*this).size();
        if(n <= d) return {};
        F ret(*this);
        ret.erase(ret.begin(), ret.begin() + d);
        return ret;
    }
    void shrink() {
        while(!(*this).empty() and (*this).back() == mint(0)) (*this).pop_back();
    }
    F rev() const {
        F ret(*this);
        reverse(begin(ret), end(ret));
        return ret;
    }
    F pre(const int deg) const {
        assert(deg >= 0);
        F ret(begin(*this), begin(*this) + min((int)(*this).size(), deg));
        if((int)ret.size() < deg) ret.resize(deg);
        return ret;
    }
    mint eval(const mint& a) const {
        const int n = (*this).size();
        mint x = 1, ret = 0;
        for(int i = 0; i < n; ++i) {
            ret += (*this)[i] * x;
            x *= a;
        }
        return ret;
    }
    void onemul(const int d, const mint& c, int deg = -1) {
        assert(deg >= -1);
        const int n = (*this).size();
        if(deg == -1) deg = n + d;
        if(deg > n) (*this).resize(deg);
        for(int i = deg - d - 1; i >= 0; --i) {
            (*this)[i + d] += (*this)[i] * c;
        }
    }
    void onediv(const int d, const mint& c, int deg = -1) {
        assert(deg >= -1);
        const int n = (*this).size();
        if(deg == -1) deg = n;
        if(deg > n) (*this).resize(deg + 1);
        for(int i = 0; i < deg - d; ++i) {
            (*this)[i + d] -= (*this)[i] * c;
        }
    }
    F diff() const {
        const int n = (*this).size();
        F ret(max(0, n - 1));
        for(int i = 1; i < n; ++i) ret[i - 1] = (*this)[i] * i;
        return ret;
    }
    F integral() const {
        const int n = (*this).size();
        static constexpr int mod = mint::mod();
        F ret(n + 1);
        ret[0] = mint(0);
        if(n > 0) ret[1] = mint(1);
        for(int i = 2; i <= n; ++i) ret[i] = (-ret[mod % i]) * (mod / i);
        for(int i = 0; i < n; ++i) ret[i + 1] *= (*this)[i];
        return ret;
    }
    F inv(int deg = -1) const {
        assert(deg >= -1);
        const int n = (*this).size();
        assert(n > 0 and (*this)[0] != mint(0));
        if(deg == -1) deg = n;
        F g(1);
        g[0] = (*this)[0].inv();
        while((int)g.size() < deg) {
            const int m = g.size();
            F f(begin(*this), begin(*this) + min(n, 2 * m));
            F r(g);
            f.resize(2 * m);
            r.resize(2 * m);
            butterfly(f);
            butterfly(r);
            for(int i = 0; i < 2 * m; ++i) f[i] *= r[i];
            butterfly_inv(f);
            f.erase(f.begin(), f.begin() + m);
            f.resize(2 * m);
            butterfly(f);
            for(int i = 0; i < 2 * m; ++i) f[i] *= r[i];
            butterfly_inv(f);
            mint in = mint(2 * m).inv();
            in *= -in;
            for(int i = 0; i < m; ++i) f[i] *= in;
            g.insert(g.end(), f.begin(), f.begin() + m);
        }
        return g.pre(deg);
    }
    F log(int deg = -1) const {
        assert(deg >= -1);
        const int n = (*this).size();
        assert(n > 0 and (*this)[0] == mint(1));
        if(deg == -1) deg = n;
        return ((*this).diff() * (*this).inv(deg)).pre(deg - 1).integral();
    }
    F exp(int deg = -1) const {
        assert(deg >= -1);
        const int n = (*this).size();
        assert(n == 0 or (*this)[0] == 0);
        if(deg == -1) deg = n;
        F Inv;
        Inv.reserve(deg + 1);
        Inv.emplace_back(mint(0));
        Inv.emplace_back(mint(1));
        auto inplace_integral = [&](F& f) -> void {
            const int n = (int)f.size();
            static constexpr int mod = mint::mod();
            while((int)Inv.size() <= n) {
                const int i = Inv.size();
                Inv.emplace_back((-Inv[mod % i]) * (mod / i));
            }
            f.insert(begin(f), mint(0));
            for(int i = 1; i <= n; ++i) f[i] *= Inv[i];
        };
        auto inplace_diff = [](F& f) -> void {
            if(f.empty()) return;
            f.erase(begin(f));
            mint coeff = 1;
            for(int i = 0; i < (int)f.size(); ++i) {
                f[i] *= coeff;
                ++coeff;
            }
        };
        F b{1, 1 < (int)(*this).size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
        for(int m = 2; m < deg; m <<= 1) {
            auto y = b;
            y.resize(2 * m);
            butterfly(y);
            z1 = z2;
            F z(m);
            for(int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
            butterfly_inv(z);
            const mint si = mint(m).inv();
            for(int i = 0; i < m; ++i) z[i] *= si;
            fill(begin(z), begin(z) + m / 2, mint(0));
            butterfly(z);
            for(int i = 0; i < m; ++i) z[i] *= -z1[i];
            butterfly_inv(z);
            for(int i = 0; i < m; ++i) z[i] *= si;
            c.insert(end(c), begin(z) + m / 2, end(z));
            z2 = c;
            z2.resize(2 * m);
            butterfly(z2);
            F x(begin((*this)), begin((*this)) + min<int>((*this).size(), m));
            x.resize(m);
            inplace_diff(x);
            x.emplace_back(mint(0));
            butterfly(x);
            for(int i = 0; i < m; ++i) x[i] *= y[i];
            butterfly_inv(x);
            for(int i = 0; i < m; ++i) x[i] *= si;
            x -= b.diff();
            x.resize(2 * m);
            for(int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0);
            butterfly(x);
            for(int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
            butterfly_inv(x);
            const mint si2 = mint(m << 1).inv();
            for(int i = 0; i < 2 * m; ++i) x[i] *= si2;
            x.pop_back();
            inplace_integral(x);
            for(int i = m; i < min<int>((*this).size(), 2 * m); ++i) x[i] += (*this)[i];
            fill(begin(x), begin(x) + m, mint(0));
            butterfly(x);
            for(int i = 0; i < 2 * m; ++i) x[i] *= y[i];
            butterfly_inv(x);
            for(int i = 0; i < 2 * m; ++i) x[i] *= si2;
            b.insert(end(b), begin(x) + m, end(x));
        }
        return b.pre(deg);
    }
    F pow(const long long k, int deg = -1) const {
        assert(deg >= -1);
        assert(k >= 0);
        const int n = (*this).size();
        if(deg == -1) deg = n;
        if(k == 0) {
            F ret(deg);
            if(deg) ret[0] = 1;
            return ret;
        }
        for(int i = 0; i < n; ++i) {
            if((*this)[i] != mint(0)) {
                const mint rev = mint(1) / (*this)[i];
                F ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
                ret *= (*this)[i].pow(k);
                ret = (ret << (i * k)).pre(deg);
                if((int)ret.size() < deg) ret.resize(deg, mint(0));
                return ret;
            }
            if(__int128_t(i + 1) * k >= deg) return F(deg, mint(0));
        }
        return F(deg, mint(0));
    }
    F shift(const mint& c) const {
        const int n = (*this).size();
        vector<mint> fact(n), ifact(n);
        fact[0] = ifact[0] = mint(1);
        for(int i = 1; i < n; ++i) fact[i] = fact[i - 1] * i;
        ifact[n - 1] = mint(1) / fact[n - 1];
        for(int i = n - 1; i > 1; --i) ifact[i - 1] = ifact[i] * i;
        F ret(*this);
        for(int i = 0; i < n; ++i) ret[i] *= fact[i];
        ret = ret.rev();
        F bs(n, mint(1));
        for(int i = 1; i < n; ++i) bs[i] = bs[i - 1] * c * ifact[i] * fact[i - 1];
        ret = (ret * bs).pre(n);
        ret = ret.rev();
        for(int i = 0; i < n; ++i) ret[i] *= ifact[i];
        return ret;
    }
};

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