#pragma once
#include <cassert>
#include <functional>
#include <type_traits>
using namespace std;
#include "formal-power-series.hpp"
#include "pow-enumerate.hpp"
// f を入力として, f(g(x)) = x を満たす g(x) mod x^{deg} を返す
template <typename mint>
FormalPowerSeries<mint> compositional_inverse(FormalPowerSeries<mint> f,
int deg = -1) {
using fps = FormalPowerSeries<mint>;
assert((int)f.size() >= 2 and f[1] != 0);
if (deg == -1) deg = f.size();
if (deg < 2) return fps{0, f[1].inverse()}.pre(deg);
int n = deg - 1;
fps h = pow_enumerate(f) * n;
for (int k = 1; k <= n; k++) h[k] /= k;
h = h.rev();
h *= h[0].inverse();
fps g = (h.log() * mint{-n}.inverse()).exp();
g *= f[1].inverse();
return (g << 1).pre(deg);
}
namespace CompositionalInverseImpl {
template <typename fps, typename F>
fps compositional_inverse_impl(F& calc_f, int deg) {
if (deg <= 2) {
fps g = std::invoke(calc_f, fps{0, 1}, 2);
assert(g[0] == 0 && g[1] != 0);
g[1] = g[1].inverse();
return g.pre(deg);
}
fps g = compositional_inverse_impl<fps>(calc_f, (deg + 1) / 2);
fps fg = std::invoke(calc_f, g, deg + 1);
fps fdg = (fg.diff() * g.diff().inv(deg)).pre(deg);
return (g - (fg - fps{0, 1}) * fdg.inv()).pre(deg);
}
} // namespace CompositionalInverseImpl
// f(g(x)) = x を満たす g(x) mod x^{deg} を返す
// calc_f(g, d) は f(g(x)) mod x^d を計算する関数
template <typename fps, typename F>
auto compositional_inverse(F&& calc_f, int deg)
-> enable_if_t<is_invocable_r_v<fps, F&, fps, int>, fps> {
return CompositionalInverseImpl::compositional_inverse_impl<fps>(calc_f,
deg);
}
template <typename fps>
fps compositional_inverse(function<fps(fps, int)> calc_f, int deg) {
return CompositionalInverseImpl::compositional_inverse_impl<fps>(calc_f,
deg);
}
/*
* @brief 逆関数
*/
#line 2 "fps/fps-compositional-inverse.hpp"
#include <cassert>
#include <functional>
#include <type_traits>
using namespace std;
#line 2 "fps/formal-power-series.hpp"
#include <algorithm>
#line 5 "fps/formal-power-series.hpp"
#include <cstdint>
#include <iterator>
#include <vector>
using namespace std;
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;
template <int N>
struct FPSBackendPriority : FPSBackendPriority<N - 1> {};
template <>
struct FPSBackendPriority<0> {};
template <typename mint>
void FormalPowerSeries<mint>::set_fft() {
fps_set_fft_impl((FormalPowerSeries<mint>*)nullptr, FPSBackendPriority<1>{});
}
template <typename mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(const FPS& r) {
if (this->empty() || r.empty()) {
this->clear();
return *this;
}
return fps_multiply_impl(*this, r, FPSBackendPriority<1>{});
}
template <typename mint>
void FormalPowerSeries<mint>::ntt() {
fps_ntt_impl(*this, FPSBackendPriority<1>{});
}
template <typename mint>
void FormalPowerSeries<mint>::intt() {
fps_intt_impl(*this, FPSBackendPriority<1>{});
}
template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
fps_ntt_doubling_impl(*this, FPSBackendPriority<1>{});
}
template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
return fps_ntt_pr_impl((FormalPowerSeries<mint>*)nullptr,
FPSBackendPriority<1>{});
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
return fps_inv_impl(*this, deg, FPSBackendPriority<1>{});
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
return fps_exp_impl(*this, deg, FPSBackendPriority<1>{});
}
/**
* @brief 多項式/形式的冪級数ライブラリ
*/
#line 2 "fps/pow-enumerate.hpp"
#line 5 "fps/pow-enumerate.hpp"
using namespace std;
#line 8 "fps/pow-enumerate.hpp"
// [x^n] f(x)^i g(x) を i=0,1,...,m で列挙
// n = (f の次数) - 1
template <typename mint>
FormalPowerSeries<mint> pow_enumerate(FormalPowerSeries<mint> f,
FormalPowerSeries<mint> g = {1},
int m = -1) {
using fps = FormalPowerSeries<mint>;
int n = f.size() - 1, k = 1;
g.resize(n + 1);
if (m == -1) m = n;
int h = 1;
while (h < n + 1) h *= 2;
fps P((n + 1) * k), Q((n + 1) * k), nP, nQ, buf, buf2;
for (int i = 0; i <= n; i++) P[i * k + 0] = g[i];
for (int i = 0; i <= n; i++) Q[i * k + 0] = -f[i];
Q[0] += 1;
while (n) {
mint inv2 = mint{2}.inverse();
mint w = mint{fps::ntt_pr()}.pow((mint::get_mod() - 1) / (2 * k));
mint iw = w.inverse();
buf2.resize(k);
auto ntt_doubling = [&]() {
copy(begin(buf), end(buf), begin(buf2));
buf2.intt();
mint c = 1;
for (int i = 0; i < k; i++) buf2[i] *= c, c *= w;
buf2.ntt();
copy(begin(buf2), end(buf2), back_inserter(buf));
};
nP.clear(), nQ.clear();
for (int i = 0; i <= n; i++) {
buf.resize(k);
copy(begin(P) + i * k, begin(P) + (i + 1) * k, begin(buf));
ntt_doubling();
copy(begin(buf), end(buf), back_inserter(nP));
buf.resize(k);
copy(begin(Q) + i * k, begin(Q) + (i + 1) * k, begin(buf));
if (i == 0) {
for (int j = 0; j < k; j++) buf[j] -= 1;
ntt_doubling();
for (int j = 0; j < k; j++) buf[j] += 1;
for (int j = 0; j < k; j++) buf[k + j] -= 1;
} else {
ntt_doubling();
}
copy(begin(buf), end(buf), back_inserter(nQ));
}
nP.resize(2 * h * 2 * k);
nQ.resize(2 * h * 2 * k);
fps p(2 * h), q(2 * h);
w = mint{fps::ntt_pr()}.pow((mint::get_mod() - 1) / (2 * h));
iw = w.inverse();
vector<int> btr;
if (n % 2) {
btr.resize(h);
for (int i = 0, lg = __builtin_ctz(h); i < h; i++) {
btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (lg - 1));
}
}
for (int j = 0; j < 2 * k; j++) {
p.assign(2 * h, 0);
q.assign(2 * h, 0);
for (int i = 0; i < h; i++) {
p[i] = nP[i * 2 * k + j], q[i] = nQ[i * 2 * k + j];
}
p.ntt(), q.ntt();
for (int i = 0; i < 2 * h; i += 2) swap(q[i], q[i + 1]);
for (int i = 0; i < 2 * h; i++) p[i] *= q[i];
for (int i = 0; i < h; i++) q[i] = q[i * 2] * q[i * 2 + 1];
if (n % 2 == 0) {
for (int i = 0; i < h; i++) p[i] = (p[i * 2] + p[i * 2 + 1]) * inv2;
} else {
mint c = inv2;
buf.resize(h);
for (int i : btr) buf[i] = (p[i * 2] - p[i * 2 + 1]) * c, c *= iw;
swap(p, buf);
}
p.resize(h), q.resize(h);
p.intt(), q.intt();
for (int i = 0; i < h; i++) nP[i * 2 * k + j] = p[i];
for (int i = 0; i < h; i++) nQ[i * 2 * k + j] = q[i];
}
nP.resize((n / 2 + 1) * 2 * k);
nQ.resize((n / 2 + 1) * 2 * k);
swap(P, nP), swap(Q, nQ);
n /= 2, h /= 2, k *= 2;
}
fps S{begin(P), begin(P) + k};
fps T{begin(Q), begin(Q) + k};
S.intt(), T.intt(), T[0] -= 1;
if (f[0] == 0) return S.rev().pre(m + 1);
return (S.rev() * (T + (fps{1} << k)).rev().inv(m + 1)).pre(m + 1);
}
/*
// 別バージョン
// [x^n] f(x)^i g(x) を i=0,1,...,m で列挙
// n = (f の次数) - 1
FormalPowerSeries<mint> pow_enumerate(FormalPowerSeries<mint> f,
FormalPowerSeries<mint> g = {1},
int m = -1) {
using fps = FormalPowerSeries<mint>;
int n = f.size() - 1, k = 1;
g.resize(n + 1);
if (m == -1) m = n;
int h = 1;
while (h < n + 1) h *= 2;
fps P(h * k), Q(h * k), nP(4 * h * k), nQ(4 * h * k), nR(2 * h * k);
for (int i = 0; i <= n; i++) P[i] = g[i], Q[i] = -f[i];
while (n) {
nP.assign(4 * h * k, 0);
nQ.assign(4 * h * k, 0);
for (int i = 0; i < k; i++) {
copy(begin(P) + i * h, begin(P) + i * h + n + 1, begin(nP) + i * 2 * h);
copy(begin(Q) + i * h, begin(Q) + i * h + n + 1, begin(nQ) + i * 2 * h);
}
nQ[k * 2 * h] += 1;
nP.ntt(), nQ.ntt();
for (int i = 0; i < 4 * h * k; i += 2) swap(nQ[i], nQ[i + 1]);
for (int i = 0; i < 4 * h * k; i++) nP[i] *= nQ[i];
for (int i = 0; i < 2 * h * k; i++) nR[i] = nQ[i * 2] * nQ[i * 2 + 1];
nP.intt(), nR.intt();
nR[0] -= 1;
P.assign(h * k, 0), Q.assign(h * k, 0);
for (int i = 0; i < 2 * k; i++) {
for (int j = 0; j <= n / 2; j++) {
P[i * h / 2 + j] = nP[i * 2 * h + j * 2 + n % 2];
Q[i * h / 2 + j] = nR[i * h + j];
}
}
n /= 2, h /= 2, k *= 2;
}
fps S{begin(P), begin(P) + k}, T{begin(Q), begin(Q) + k};
T.push_back(1);
return (S.rev() * T.rev().inv(m + 1)).pre(m + 1);
}
*/
/**
* @brief pow 列挙
*/
#line 10 "fps/fps-compositional-inverse.hpp"
// f を入力として, f(g(x)) = x を満たす g(x) mod x^{deg} を返す
template <typename mint>
FormalPowerSeries<mint> compositional_inverse(FormalPowerSeries<mint> f,
int deg = -1) {
using fps = FormalPowerSeries<mint>;
assert((int)f.size() >= 2 and f[1] != 0);
if (deg == -1) deg = f.size();
if (deg < 2) return fps{0, f[1].inverse()}.pre(deg);
int n = deg - 1;
fps h = pow_enumerate(f) * n;
for (int k = 1; k <= n; k++) h[k] /= k;
h = h.rev();
h *= h[0].inverse();
fps g = (h.log() * mint{-n}.inverse()).exp();
g *= f[1].inverse();
return (g << 1).pre(deg);
}
namespace CompositionalInverseImpl {
template <typename fps, typename F>
fps compositional_inverse_impl(F& calc_f, int deg) {
if (deg <= 2) {
fps g = std::invoke(calc_f, fps{0, 1}, 2);
assert(g[0] == 0 && g[1] != 0);
g[1] = g[1].inverse();
return g.pre(deg);
}
fps g = compositional_inverse_impl<fps>(calc_f, (deg + 1) / 2);
fps fg = std::invoke(calc_f, g, deg + 1);
fps fdg = (fg.diff() * g.diff().inv(deg)).pre(deg);
return (g - (fg - fps{0, 1}) * fdg.inv()).pre(deg);
}
} // namespace CompositionalInverseImpl
// f(g(x)) = x を満たす g(x) mod x^{deg} を返す
// calc_f(g, d) は f(g(x)) mod x^d を計算する関数
template <typename fps, typename F>
auto compositional_inverse(F&& calc_f, int deg)
-> enable_if_t<is_invocable_r_v<fps, F&, fps, int>, fps> {
return CompositionalInverseImpl::compositional_inverse_impl<fps>(calc_f,
deg);
}
template <typename fps>
fps compositional_inverse(function<fps(fps, int)> calc_f, int deg) {
return CompositionalInverseImpl::compositional_inverse_impl<fps>(calc_f,
deg);
}
/*
* @brief 逆関数
*/
逆関数