fps2.hpp

[詳解]

template <typename mint>
struct NTT {
    static constexpr uint32_t get_pr() {
        uint32_t _mod = mint::get_mod();
        using u64 = uint64_t;
        u64 ds[32] = {};
        int idx = 0;
        u64 m = _mod - 1;
        for (u64 i = 2; i * i <= m; ++i) {
            if (m % i == 0) {
                ds[idx++] = i;
                while (m % i == 0) m /= i;
            }
        }
        if (m != 1) ds[idx++] = m;
 
        uint32_t _pr = 2;
        while (1) {
            int flg = 1;
            for (int i = 0; i < idx; ++i) {
                u64 a = _pr, b = (_mod - 1) / ds[i], r = 1;
                while (b) {
                    if (b & 1) r = r * a % _mod;
                    a = a * a % _mod;
                    b >>= 1;
                }
                if (r == 1) {
                    flg = 0;
                    break;
                }
            }
            if (flg == 1) break;
            ++_pr;
        }
        return _pr;
    };
 
    static constexpr uint32_t mod = mint::get_mod();
    static constexpr uint32_t pr = get_pr();
    static constexpr int level = __builtin_ctzll(mod - 1);
    mint dw[level], dy[level];
 
    void setwy(int k) {
        mint w[level], y[level];
        w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
        y[k - 1] = w[k - 1].inverse();
        for (int i = k - 2; i > 0; --i)
            w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
        dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
        for (int i = 3; i < k; ++i) {
            dw[i] = dw[i - 1] * y[i - 2] * w[i];
            dy[i] = dy[i - 1] * w[i - 2] * y[i];
        }
    }
 
    NTT() { setwy(level); }
 
    void fft4(vector<mint> &a, int k) {
        if ((int)a.size() <= 1) return;
        if (k == 1) {
            mint a1 = a[1];
            a[1] = a[0] - a[1];
            a[0] = a[0] + a1;
            return;
        }
        if (k & 1) {
            int v = 1 << (k - 1);
            for (int j = 0; j < v; ++j) {
                mint ajv = a[j + v];
                a[j + v] = a[j] - ajv;
                a[j] += ajv;
            }
        }
        int u = 1 << (2 + (k & 1));
        int v = 1 << (k - 2 - (k & 1));
        mint one = mint(1);
        mint imag = dw[1];
        while (v) {
            // jh = 0
            {
                int j0 = 0;
                int j1 = v;
                int j2 = j1 + v;
                int j3 = j2 + v;
                for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
                    mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
                    mint t0p2 = t0 + t2, t1p3 = t1 + t3;
                    mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
                    a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
                    a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
                }
            }
            // jh >= 1
            mint ww = one, xx = one * dw[2], wx = one;
            for (int jh = 4; jh < u;) {
                ww = xx * xx, wx = ww * xx;
                int j0 = jh * v;
                int je = j0 + v;
                int j2 = je + v;
                for (; j0 < je; ++j0, ++j2) {
                    mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww,
                         t3 = a[j2 + v] * wx;
                    mint t0p2 = t0 + t2, t1p3 = t1 + t3;
                    mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
                    a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3;
                    a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3;
                }
                xx *= dw[__builtin_ctzll((jh += 4))];
            }
            u <<= 2;
            v >>= 2;
        }
    }
 
    void ifft4(vector<mint> &a, int k) {
        if ((int)a.size() <= 1) return;
        if (k == 1) {
            mint a1 = a[1];
            a[1] = a[0] - a[1];
            a[0] = a[0] + a1;
            return;
        }
        int u = 1 << (k - 2);
        int v = 1;
        mint one = mint(1);
        mint imag = dy[1];
        while (u) {
            // jh = 0
            {
                int j0 = 0;
                int j1 = v;
                int j2 = v + v;
                int j3 = j2 + v;
                for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
                    mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
                    mint t0p1 = t0 + t1, t2p3 = t2 + t3;
                    mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag;
                    a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
                    a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
                }
            }
            // jh >= 1
            mint ww = one, xx = one * dy[2], yy = one;
            u <<= 2;
            for (int jh = 4; jh < u;) {
                ww = xx * xx, yy = xx * imag;
                int j0 = jh * v;
                int je = j0 + v;
                int j2 = je + v;
                for (; j0 < je; ++j0, ++j2) {
                    mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
                    mint t0p1 = t0 + t1, t2p3 = t2 + t3;
                    mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
                    a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
                    a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww;
                }
                xx *= dy[__builtin_ctzll(jh += 4)];
            }
            u >>= 4;
            v <<= 2;
        }
        if (k & 1) {
            u = 1 << (k - 1);
            for (int j = 0; j < u; ++j) {
                mint ajv = a[j] - a[j + u];
                a[j] += a[j + u];
                a[j + u] = ajv;
            }
        }
    }
 
    void ntt(vector<mint> &a) {
        if ((int)a.size() <= 1) return;
        fft4(a, __builtin_ctz(a.size()));
    }
 
    void intt(vector<mint> &a) {
        if ((int)a.size() <= 1) return;
        ifft4(a, __builtin_ctz(a.size()));
        mint iv = mint(a.size()).inverse();
        for (auto &x : a) x *= iv;
    }
 
    vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
        int l = a.size() + b.size() - 1;
        if (min<int>(a.size(), b.size()) <= 40) {
            vector<mint> s(l);
            for (int i = 0; i < (int)a.size(); ++i)
                for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j];
            return s;
        }
        int k = 2, M = 4;
        while (M < l) M <<= 1, ++k;
        setwy(k);
        vector<mint> s(M);
        for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i];
        fft4(s, k);
        if (a.size() == b.size() && a == b) {
            for (int i = 0; i < M; ++i) s[i] *= s[i];
        } else {
            vector<mint> t(M);
            for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i];
            fft4(t, k);
            for (int i = 0; i < M; ++i) s[i] *= t[i];
        }
        ifft4(s, k);
        s.resize(l);
        mint invm = mint(M).inverse();
        for (int i = 0; i < l; ++i) s[i] *= invm;
        return s;
    }
 
    void ntt_doubling(vector<mint> &a) {
        int M = (int)a.size();
        auto b = a;
        intt(b);
        mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
        for (int i = 0; i < M; i++) b[i] *= r, r *= zeta;
        ntt(b);
        copy(begin(b), end(b), back_inserter(a));
    }
};
 
template <typename mint>
struct FormalPowerSeries : vector<mint> {
    using vector<mint>::vector;
    using FPS = FormalPowerSeries;
 
    FPS &operator+=(const FPS &r) {
        if (r.size() > this->size()) this->resize(r.size());
        for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
        return *this;
    }
 
    FPS &operator+=(const mint &r) {
        if (this->empty()) this->resize(1);
        (*this)[0] += r;
        return *this;
    }
 
    FPS &operator-=(const FPS &r) {
        if (r.size() > this->size()) this->resize(r.size());
        for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
        return *this;
    }
 
    FPS &operator-=(const mint &r) {
        if (this->empty()) this->resize(1);
        (*this)[0] -= r;
        return *this;
    }
 
    FPS &operator*=(const mint &v) {
        for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
        return *this;
    }
 
    FPS &operator/=(const FPS &r) {
        if (this->size() < r.size()) {
            this->clear();
            return *this;
        }
        int n = this->size() - r.size() + 1;
        if ((int)r.size() <= 64) {
            FPS f(*this), g(r);
            g.shrink();
            mint coeff = g.back().inverse();
            for (auto &x : g) x *= coeff;
            int deg = (int)f.size() - (int)g.size() + 1;
            int gs = g.size();
            FPS quo(deg);
            for (int i = deg - 1; i >= 0; i--) {
                quo[i] = f[i + gs - 1];
                for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
            }
            *this = quo * coeff;
            this->resize(n, mint(0));
            return *this;
        }
        return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
    }
 
    FPS &operator%=(const FPS &r) {
        *this -= *this / r * r;
        shrink();
        return *this;
    }
 
    FPS operator+(const FPS &r) const { return FPS(*this) += r; }
    FPS operator+(const mint &v) const { return FPS(*this) += v; }
    FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
    FPS operator-(const mint &v) const { return FPS(*this) -= v; }
    FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
    FPS operator*(const mint &v) const { return FPS(*this) *= v; }
    FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
    FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
    FPS operator-() const {
        FPS ret(this->size());
        for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
        return ret;
    }
 
    void shrink() {
        while (this->size() && this->back() == mint(0)) this->pop_back();
    }
 
    FPS rev() const {
        FPS ret(*this);
        reverse(begin(ret), end(ret));
        return ret;
    }
 
    FPS dot(FPS r) const {
        FPS ret(min(this->size(), r.size()));
        for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
        return ret;
    }
 
    // 前 sz 項を取ってくる。sz に足りない項は 0 埋めする
    FPS pre(int sz) const {
        FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz));
        if ((int)ret.size() < sz) ret.resize(sz);
        return ret;
    }
 
    FPS operator>>(int sz) const {
        if ((int)this->size() <= sz) return {};
        FPS ret(*this);
        ret.erase(ret.begin(), ret.begin() + sz);
        return ret;
    }
 
    FPS operator<<(int sz) const {
        FPS ret(*this);
        ret.insert(ret.begin(), sz, mint(0));
        return ret;
    }
 
    FPS diff() const {
        const int n = (int)this->size();
        FPS ret(max(0, n - 1));
        mint one(1), coeff(1);
        for (int i = 1; i < n; i++) {
            ret[i - 1] = (*this)[i] * coeff;
            coeff += one;
        }
        return ret;
    }
 
    FPS integral() const {
        const int n = (int)this->size();
        FPS ret(n + 1);
        ret[0] = mint(0);
        if (n > 0) ret[1] = mint(1);
        auto mod = mint::get_mod();
        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;
    }
 
    mint eval(mint x) const {
        mint r = 0, w = 1;
        for (auto &v : *this) r += w * v, w *= x;
        return r;
    }
 
    FPS log(int deg = -1) const {
        assert(!(*this).empty() && (*this)[0] == mint(1));
        if (deg == -1) deg = (int)this->size();
        return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
    }
 
    FPS pow(int64_t k, int deg = -1) const {
        const int n = (int)this->size();
        if (deg == -1) deg = n;
        if (k == 0) {
            FPS ret(deg);
            if (deg) ret[0] = 1;
            return ret;
        }
        for (int i = 0; i < n; i++) {
            if ((*this)[i] != mint(0)) {
                mint rev = mint(1) / (*this)[i];
                FPS 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 FPS(deg, mint(0));
        }
        return FPS(deg, mint(0));
    }
 
    static void *ntt_ptr;
    static void set_fft();
    FPS &operator*=(const FPS &r);
    void ntt();
    void intt();
    void ntt_doubling();
    static int ntt_pr();
    FPS inv(int deg = -1) const;
    FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
 
/**
 * @brief 多項式/形式的冪級数ライブラリ
 * @docs docs/fps/formal-power-series.md
 */
#line 8 "fps/sparse-fps.hpp"
 
// g が sparse を仮定, f * g.inv() を計算
template <typename mint>
FormalPowerSeries<mint> sparse_div(const FormalPowerSeries<mint>& f,
        const FormalPowerSeries<mint>& g,
        int deg = -1) {
    assert(g.empty() == false && g[0] != mint(0));
    if (deg == -1) deg = f.size();
    mint ig0 = g[0].inverse();
    FormalPowerSeries<mint> s = f * ig0;
    s.resize(deg);
    vector<pair<int, mint>> gs;
    for (int i = 1; i < (int)g.size(); i++) {
        if (g[i] != 0) gs.emplace_back(i, g[i] * ig0);
    }
    for (int i = 0; i < deg; i++) {
        for (auto& [j, g_j] : gs) {
            if (i + j >= deg) break;
            s[i + j] -= s[i] * g_j;
        }
    }
    return s;
}
 
template <typename mint>
FormalPowerSeries<mint> sparse_inv(const FormalPowerSeries<mint>& f,
        int deg = -1) {
    assert(f.empty() == false && f[0] != mint(0));
    if (deg == -1) deg = f.size();
    vector<pair<int, mint>> fs;
    for (int i = 1; i < (int)f.size(); i++) {
        if (f[i] != 0) fs.emplace_back(i, f[i]);
    }
    FormalPowerSeries<mint> g(deg);
    mint if0 = f[0].inverse();
    if (0 < deg) g[0] = if0;
    for (int k = 1; k < deg; k++) {
        for (auto& [j, fj] : fs) {
            if (k < j) break;
            g[k] += g[k - j] * fj;
        }
        g[k] *= -if0;
    }
    return g;
}
 
template <typename mint>
FormalPowerSeries<mint> sparse_log(const FormalPowerSeries<mint>& f,
        int deg = -1) {
    assert(f.empty() == false && f[0] == 1);
    if (deg == -1) deg = f.size();
    vector<pair<int, mint>> fs;
    for (int i = 1; i < (int)f.size(); i++) {
        if (f[i] != 0) fs.emplace_back(i, f[i]);
    }
 
    int mod = mint::get_mod();
    static vector<mint> invs{1, 1};
    while ((int)invs.size() <= deg) {
        int i = invs.size();
        invs.push_back((-invs[mod % i]) * (mod / i));
    }
 
    FormalPowerSeries<mint> g(deg);
    for (int k = 0; k < deg - 1; k++) {
        for (auto& [j, fj] : fs) {
            if (k < j) break;
            int i = k - j;
            g[k + 1] -= g[i + 1] * fj * (i + 1);
        }
        g[k + 1] *= invs[k + 1];
        if (k + 1 < (int)f.size()) g[k + 1] += f[k + 1];
    }
    return g;
}
 
template <typename mint>
FormalPowerSeries<mint> sparse_exp(const FormalPowerSeries<mint>& f,
        int deg = -1) {
    assert(f.empty() or f[0] == 0);
    if (deg == -1) deg = f.size();
    vector<pair<int, mint>> fs;
    for (int i = 1; i < (int)f.size(); i++) {
        if (f[i] != 0) fs.emplace_back(i, f[i]);
    }
 
    int mod = mint::get_mod();
    static vector<mint> invs{1, 1};
    while ((int)invs.size() <= deg) {
        int i = invs.size();
        invs.push_back((-invs[mod % i]) * (mod / i));
    }
 
    FormalPowerSeries<mint> g(deg);
    if (deg) g[0] = 1;
    for (int k = 0; k < deg - 1; k++) {
        for (auto& [ip1, fip1] : fs) {
            int i = ip1 - 1;
            if (k < i) break;
            g[k + 1] += fip1 * g[k - i] * (i + 1);
        }
        g[k + 1] *= invs[k + 1];
    }
    return g;
}
 
template <typename mint>
FormalPowerSeries<mint> sparse_pow(const FormalPowerSeries<mint>& f,
        long long k, int deg = -1) {
    if (deg == -1) deg = f.size();
    if (k == 0) {
        FormalPowerSeries<mint> g(deg);
        if (deg) g[0] = 1;
        return g;
    }
    int zero = 0;
    while (zero != (int)f.size() and f[zero] == 0) zero++;
    if (zero == (int)f.size() or __int128_t(zero) * k >= deg) {
        return FormalPowerSeries<mint>(deg, 0);
    }
    if (zero != 0) {
        FormalPowerSeries<mint> suf{begin(f) + zero, end(f)};
        auto g = sparse_pow(suf, k, deg - zero * k);
        FormalPowerSeries<mint> h(zero * k, 0);
        copy(begin(g), end(g), back_inserter(h));
        return h;
    }
 
    int mod = mint::get_mod();
    static vector<mint> invs{1, 1};
    while ((int)invs.size() <= deg) {
        int i = invs.size();
        invs.push_back((-invs[mod % i]) * (mod / i));
    }
 
    vector<pair<int, mint>> fs;
    for (int i = 1; i < (int)f.size(); i++) {
        if (f[i] != 0) fs.emplace_back(i, f[i]);
    }
 
    FormalPowerSeries<mint> g(deg);
    g[0] = f[0].pow(k);
    mint denom = f[0].inverse();
    k %= mint::get_mod();
    for (int a = 1; a < deg; a++) {
        for (auto& [i, f_i] : fs) {
            if (a < i) break;
            g[a] += f_i * g[a - i] * ((k + 1) * i - a);
        }
        g[a] *= denom * invs[a];
    }
    return g;
}
 
template <typename mint>
void FormalPowerSeries<mint>::set_fft() {
    if (!ntt_ptr) ntt_ptr = new NTT<mint>;
}
 
template <typename mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(
        const FormalPowerSeries<mint>& r) {
    if (this->empty() || r.empty()) {
        this->clear();
        return *this;
    }
    set_fft();
    auto ret = static_cast<NTT<mint>*>(ntt_ptr)->multiply(*this, r);
    return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
 
template <typename mint>
void FormalPowerSeries<mint>::ntt() {
    set_fft();
    static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}
 
template <typename mint>
void FormalPowerSeries<mint>::intt() {
    set_fft();
    static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}
 
template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
    set_fft();
    static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}
 
template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
    set_fft();
    return static_cast<NTT<mint>*>(ntt_ptr)->pr;
}
 
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
    assert((*this)[0] != mint(0));
    if (deg == -1) deg = (int)this->size();
    FormalPowerSeries<mint> res(deg);
    res[0] = {mint(1) / (*this)[0]};
    for (int d = 1; d < deg; d <<= 1) {
        FormalPowerSeries<mint> 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];
        f.ntt();
        g.ntt();
        for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
        f.intt();
        for (int j = 0; j < d; j++) f[j] = 0;
        f.ntt();
        for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
        f.intt();
        for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
    }
    return res.pre(deg);
}
 
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
    using fps = FormalPowerSeries<mint>;
    assert((*this).size() == 0 || (*this)[0] == mint(0));
    if (deg == -1) deg = this->size();
 
    fps inv;
    inv.reserve(deg + 1);
    inv.push_back(mint(0));
    inv.push_back(mint(1));
 
    auto inplace_integral = [&](fps& F) -> void {
        const int n = (int)F.size();
        auto mod = mint::get_mod();
        while ((int)inv.size() <= n) {
            int i = inv.size();
            inv.push_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 = [](fps& F) -> void {
        if (F.empty()) return;
        F.erase(begin(F));
        mint coeff = 1, one = 1;
        for (int i = 0; i < (int)F.size(); i++) {
            F[i] *= coeff;
            coeff += one;
        }
    };
 
    fps b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1};
    for (int m = 2; m < deg; m *= 2) {
        auto y = b;
        y.resize(2 * m);
        y.ntt();
        z1 = z2;
        fps z(m);
        for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
        z.intt();
        fill(begin(z), begin(z) + m / 2, mint(0));
        z.ntt();
        for (int i = 0; i < m; ++i) z[i] *= -z1[i];
        z.intt();
        c.insert(end(c), begin(z) + m / 2, end(z));
        z2 = c;
        z2.resize(2 * m);
        z2.ntt();
        fps x(begin(*this), begin(*this) + min<int>(this->size(), m));
        x.resize(m);
        inplace_diff(x);
        x.push_back(mint(0));
        x.ntt();
        for (int i = 0; i < m; ++i) x[i] *= y[i];
        x.intt();
        x -= b.diff();
        x.resize(2 * m);
        for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0);
        x.ntt();
        for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
        x.intt();
        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));
        x.ntt();
        for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
        x.intt();
        b.insert(end(b), begin(x) + m, end(x));
    }
    return fps{begin(b), begin(b) + deg};
}
 
 
 
template <typename mint>
FormalPowerSeries<mint> sqrt(const FormalPowerSeries<mint> &f, int deg = -1) {
    if (deg == -1) deg = (int)f.size();
    if ((int)f.size() == 0) return FormalPowerSeries<mint>(deg, 0);
    if (f[0] == mint(0)) {
        for (int i = 1; i < (int)f.size(); i++) {
            if (f[i] != mint(0)) {
                if (i & 1) return {};
                if (deg - i / 2 <= 0) break;
                auto ret = sqrt(f >> i, deg - i / 2);
                if (ret.empty()) return {};
                ret = ret << (i / 2);
                if ((int)ret.size() < deg) ret.resize(deg, mint(0));
                return ret;
            }
        }
        return FormalPowerSeries<mint>(deg, 0);
    }
 
    int64_t sqr = mod_sqrt(f[0].get(), mint::get_mod());
    if (sqr == -1) return {};
    assert(sqr * sqr % mint::get_mod() == f[0].get());
    FormalPowerSeries<mint> ret = {mint(sqr)};
    mint inv2 = mint(2).inverse();
    for (int i = 1; i < deg; i <<= 1) {
        ret = (ret + f.pre(i << 1) * ret.inv(i << 1)) * inv2;
    }
    return ret.pre(deg);
}
 
template <typename mint>
pair<FormalPowerSeries<mint>, FormalPowerSeries<mint>> circular(
        const FormalPowerSeries<mint> &fre, const FormalPowerSeries<mint> &fim,
        int deg = -1) {
    using fps = FormalPowerSeries<mint>;
    assert(fre.size() == 0 || fre[0] == mint(0));
    assert(fim.size() == 0 || fim[0] == mint(0));
    if (deg == -1) deg = (int)max(fre.size(), fim.size());
    fps re({mint(1)}), im({mint(0)});
 
    fps::set_fft();
    if (fps::ntt_ptr == nullptr) {
        for (int i = 1; i < deg; i <<= 1) {
            fps dre = re.diff();
            fps dim = im.diff();
            fps fhypot = (re * re + im * im).inv(i << 1);
            fps ere = dre * re + dim * im;
            fps eim = dim * re - dre * im;
            fps logre = (ere * fhypot).pre((i << 1) - 1).integral();
            fps logim = (eim * fhypot).pre((i << 1) - 1).integral();
            fps gre = (-logre) + mint(1) - fim.pre(i << 1);
            fps gim = (-logim) + fre.pre(i << 1);
            fps hre = (re * gre - im * gim).pre(i << 1);
            fps him = (re * gim + im * gre).pre(i << 1);
            swap(re, hre);
            swap(im, him);
        }
    } else {
        for (int i = 1; i < deg; i <<= 1) {
            fps dre = re.diff();
            fps dim = im.diff();
            re.resize(i << 1);
            im.resize(i << 1);
            dre.resize(i << 1);
            dim.resize(i << 1);
            re.ntt();
            im.ntt();
            dre.ntt();
            dim.ntt();
            fps fhypot(i << 1), ere(i << 1), eim(i << 1);
            for (int j = 0; j < 2 * i; j++) {
                fhypot[j] = re[j] * re[j] + im[j] * im[j];
                ere[j] = dre[j] * re[j] + dim[j] * im[j];
                eim[j] = dim[j] * re[j] - dre[j] * im[j];
            }
            fhypot.intt();
            fhypot = fhypot.inv(i << 1);
            fhypot.resize(i << 2);
            fhypot.ntt();
            ere.ntt_doubling();
            eim.ntt_doubling();
            fps logre(i << 2), logim(i << 2);
            for (int j = 0; j < 4 * i; j++) {
                logre[j] = ere[j] * fhypot[j];
                logim[j] = eim[j] * fhypot[j];
            }
            logre.intt();
            logim.intt();
            logre = logre.pre((i << 1) - 1).integral();
            logim = logim.pre((i << 1) - 1).integral();
            fps gre = (-logre) + mint(1) - fim.pre(i << 1);
            fps gim = (-logim) + fre.pre(i << 1);
            gre.resize(i << 2);
            gim.resize(i << 2);
            gre.ntt();
            gim.ntt();
            re.ntt_doubling();
            im.ntt_doubling();
            fps hre(i << 2), him(i << 2);
            for (int j = 0; j < 4 * i; j++) {
                hre[j] = re[j] * gre[j] - im[j] * gim[j];
                him[j] = re[j] * gim[j] + im[j] * gre[j];
            }
            hre.intt();
            him.intt();
            hre = hre.pre(i << 1);
            him = him.pre(i << 1);
            swap(re, hre);
            swap(im, him);
        }
    }
    return make_pair(re.pre(deg), im.pre(deg));
}
 
// calculate F(x + a)
template <typename mint>
FormalPowerSeries<mint> TaylorShift(FormalPowerSeries<mint> f, mint a,
        Binomial<mint>& C) {
    using fps = FormalPowerSeries<mint>;
    int N = f.size();
    for (int i = 0; i < N; i++) f[i] *= C.fac(i);
    reverse(begin(f), end(f));
    fps g(N, mint(1));
    for (int i = 1; i < N; i++) g[i] = g[i - 1] * a * C.inv(i);
    f = (f * g).pre(N);
    reverse(begin(f), end(f));
    for (int i = 0; i < N; i++) f[i] *= C.finv(i);
    return f;
}