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#include "math/fps/formal-power-series.hpp"
#pragma once
#include "../fft/arbitrary-mod-convolution.hpp"
/**
* @brief Formal Power Series(形式的冪級数)
*/
template <typename T>
struct FormalPowerSeries : vector<T> {
using vector<T>::vector;
using P = FormalPowerSeries;
using Conv = ArbitraryModConvolution<T>;
P pre(int deg) const {
return P(begin(*this), begin(*this) + min((int)this->size(), deg));
}
P rev(int deg = -1) const {
P ret(*this);
if (deg != -1) ret.resize(deg, T(0));
reverse(begin(ret), end(ret));
return ret;
}
void shrink() {
while (this->size() && this->back() == T(0)) this->pop_back();
}
P operator+(const P &r) const { return P(*this) += r; }
P operator+(const T &v) const { return P(*this) += v; }
P operator-(const P &r) const { return P(*this) -= r; }
P operator-(const T &v) const { return P(*this) -= v; }
P operator*(const P &r) const { return P(*this) *= r; }
P operator*(const T &v) const { return P(*this) *= v; }
P operator/(const P &r) const { return P(*this) /= r; }
P operator%(const P &r) const { return P(*this) %= r; }
P &operator+=(const P &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < r.size(); i++) (*this)[i] += r[i];
return *this;
}
P &operator-=(const P &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < r.size(); i++) (*this)[i] -= r[i];
return *this;
}
// https://judge.yosupo.jp/problem/convolution_mod
P &operator*=(const P &r) {
if (this->empty() || r.empty()) {
this->clear();
return *this;
}
auto ret = Conv::multiply(*this, r);
return *this = {begin(ret), end(ret)};
}
P &operator/=(const P &r) {
if (this->size() < r.size()) {
this->clear();
return *this;
}
int n = this->size() - r.size() + 1;
return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
}
P &operator%=(const P &r) { return *this -= *this / r * r; }
// https://judge.yosupo.jp/problem/division_of_polynomials
pair<P, P> div_mod(const P &r) {
P q = *this / r;
return make_pair(q, *this - q * r);
}
P operator-() const {
P ret(this->size());
for (int i = 0; i < this->size(); i++) ret[i] = -(*this)[i];
return ret;
}
P &operator+=(const T &r) {
if (this->empty()) this->resize(1);
(*this)[0] += r;
return *this;
}
P &operator-=(const T &r) {
if (this->empty()) this->resize(1);
(*this)[0] -= r;
return *this;
}
P &operator*=(const T &v) {
for (int i = 0; i < this->size(); i++) (*this)[i] *= v;
return *this;
}
P dot(P r) const {
P ret(min(this->size(), r.size()));
for (int i = 0; i < ret.size(); i++) ret[i] = (*this)[i] * r[i];
return ret;
}
P operator>>(int sz) const {
if (this->size() <= sz) return {};
P ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
P operator<<(int sz) const {
P ret(*this);
ret.insert(ret.begin(), sz, T(0));
return ret;
}
T operator()(T x) const {
T r = 0, w = 1;
for (auto &v : *this) {
r += w * v;
w *= x;
}
return r;
}
P diff() const {
const int n = (int)this->size();
P ret(max(0, n - 1));
for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i);
return ret;
}
P integral() const {
const int n = (int)this->size();
P ret(n + 1);
ret[0] = T(0);
for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
return ret;
}
// https://judge.yosupo.jp/problem/inv_of_formal_power_series
// F(0) must not be 0
P inv(int deg = -1) const {
assert(((*this)[0]) != T(0));
const int n = (int)this->size();
if (deg == -1) deg = n;
P ret({T(1) / (*this)[0]});
for (int i = 1; i < deg; i <<= 1) {
ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1);
}
return ret.pre(deg);
}
// https://judge.yosupo.jp/problem/log_of_formal_power_series
// F(0) must be 1
P log(int deg = -1) const {
assert((*this)[0] == T(1));
const int n = (int)this->size();
if (deg == -1) deg = n;
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
// https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
P sqrt(
int deg = -1,
const function<T(T)> &get_sqrt = [](T) { return T(1); }) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
if ((*this)[0] == T(0)) {
for (int i = 1; i < n; i++) {
if ((*this)[i] != T(0)) {
if (i & 1) return {};
if (deg - i / 2 <= 0) break;
auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt);
if (ret.empty()) return {};
ret = ret << (i / 2);
if (ret.size() < deg) ret.resize(deg, T(0));
return ret;
}
}
return P(deg, 0);
}
auto sqr = T(get_sqrt((*this)[0]));
if (sqr * sqr != (*this)[0]) return {};
P ret{sqr};
T inv2 = T(1) / T(2);
for (int i = 1; i < deg; i <<= 1) {
ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
}
return ret.pre(deg);
}
P sqrt(const function<T(T)> &get_sqrt, int deg = -1) const {
return sqrt(deg, get_sqrt);
}
// https://judge.yosupo.jp/problem/exp_of_formal_power_series
// F(0) must be 0
P exp(int deg = -1) const {
if (deg == -1) deg = this->size();
assert((*this)[0] == T(0));
const int n = (int)this->size();
if (deg == -1) deg = n;
P ret({T(1)});
for (int i = 1; i < deg; i <<= 1) {
ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1);
}
return ret.pre(deg);
}
// https://judge.yosupo.jp/problem/pow_of_formal_power_series
P pow(int64_t k, int deg = -1) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
if (k == 0) {
P ret(deg, T(0));
ret[0] = T(1);
return ret;
}
for (int i = 0; i < n; i++) {
if (i * k > deg) return P(deg, T(0));
if ((*this)[i] != T(0)) {
T rev = T(1) / (*this)[i];
P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
ret = (ret << (i * k)).pre(deg);
if (ret.size() < deg) ret.resize(deg, T(0));
return ret;
}
}
return *this;
}
// https://yukicoder.me/problems/no/215
P mod_pow(int64_t k, P g) const {
P modinv = g.rev().inv();
auto get_div = [&](P base) {
if (base.size() < g.size()) {
base.clear();
return base;
}
int n = base.size() - g.size() + 1;
return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
};
P x(*this), ret{1};
while (k > 0) {
if (k & 1) {
ret *= x;
ret -= get_div(ret) * g;
ret.shrink();
}
x *= x;
x -= get_div(x) * g;
x.shrink();
k >>= 1;
}
return ret;
}
// https://judge.yosupo.jp/problem/polynomial_taylor_shift
P taylor_shift(T c) const {
int n = (int)this->size();
vector<T> fact(n), rfact(n);
fact[0] = rfact[0] = T(1);
for (int i = 1; i < n; i++) fact[i] = fact[i - 1] * T(i);
rfact[n - 1] = T(1) / fact[n - 1];
for (int i = n - 1; i > 1; i--) rfact[i - 1] = rfact[i] * T(i);
P p(*this);
for (int i = 0; i < n; i++) p[i] *= fact[i];
p = p.rev();
P bs(n, T(1));
for (int i = 1; i < n; i++) bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
p = (p * bs).pre(n);
p = p.rev();
for (int i = 0; i < n; i++) p[i] *= rfact[i];
return p;
}
};
template <typename Mint>
using FPS = FormalPowerSeries<Mint>;
#line 2 "math/fps/formal-power-series.hpp"
#line 1 "math/fft/fast-fourier-transform.hpp"
namespace FastFourierTransform {
using real = double;
struct C {
real x, y;
C() : x(0), y(0) {}
C(real x, real y) : x(x), y(y) {}
inline C operator+(const C &c) const { return C(x + c.x, y + c.y); }
inline C operator-(const C &c) const { return C(x - c.x, y - c.y); }
inline C operator*(const C &c) const {
return C(x * c.x - y * c.y, x * c.y + y * c.x);
}
inline C conj() const { return C(x, -y); }
};
const real PI = acosl(-1);
int base = 1;
vector<C> rts = {{0, 0}, {1, 0}};
vector<int> rev = {0, 1};
void ensure_base(int nbase) {
if (nbase <= base) return;
rev.resize(1 << nbase);
rts.resize(1 << nbase);
for (int i = 0; i < (1 << nbase); i++) {
rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1));
}
while (base < nbase) {
real angle = PI * 2.0 / (1 << (base + 1));
for (int i = 1 << (base - 1); i < (1 << base); i++) {
rts[i << 1] = rts[i];
real angle_i = angle * (2 * i + 1 - (1 << base));
rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i));
}
++base;
}
}
void fft(vector<C> &a, int n) {
assert((n & (n - 1)) == 0);
int zeros = __builtin_ctz(n);
ensure_base(zeros);
int shift = base - zeros;
for (int i = 0; i < n; i++) {
if (i < (rev[i] >> shift)) {
swap(a[i], a[rev[i] >> shift]);
}
}
for (int k = 1; k < n; k <<= 1) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
C z = a[i + j + k] * rts[j + k];
a[i + j + k] = a[i + j] - z;
a[i + j] = a[i + j] + z;
}
}
}
}
vector<int64_t> multiply(const vector<int> &a, const vector<int> &b) {
int need = (int)a.size() + (int)b.size() - 1;
int nbase = 1;
while ((1 << nbase) < need) nbase++;
ensure_base(nbase);
int sz = 1 << nbase;
vector<C> fa(sz);
for (int i = 0; i < sz; i++) {
int x = (i < (int)a.size() ? a[i] : 0);
int y = (i < (int)b.size() ? b[i] : 0);
fa[i] = C(x, y);
}
fft(fa, sz);
C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0);
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r;
fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r;
fa[i] = z;
}
for (int i = 0; i < (sz >> 1); i++) {
C A0 = (fa[i] + fa[i + (sz >> 1)]) * t;
C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * rts[(sz >> 1) + i];
fa[i] = A0 + A1 * s;
}
fft(fa, sz >> 1);
vector<int64_t> ret(need);
for (int i = 0; i < need; i++) {
ret[i] = llround(i & 1 ? fa[i >> 1].y : fa[i >> 1].x);
}
return ret;
}
}; // namespace FastFourierTransform
#line 2 "math/fft/arbitrary-mod-convolution.hpp"
/*
* @brief Arbitrary Mod Convolution(任意mod畳み込み)
*/
template <typename T>
struct ArbitraryModConvolution {
using real = FastFourierTransform::real;
using C = FastFourierTransform::C;
ArbitraryModConvolution() = default;
static vector<T> multiply(const vector<T> &a, const vector<T> &b,
int need = -1) {
if (need == -1) need = a.size() + b.size() - 1;
int nbase = 0;
while ((1 << nbase) < need) nbase++;
FastFourierTransform::ensure_base(nbase);
int sz = 1 << nbase;
vector<C> fa(sz);
for (int i = 0; i < a.size(); i++) {
fa[i] = C(a[i].val() & ((1 << 15) - 1), a[i].val() >> 15);
}
fft(fa, sz);
vector<C> fb(sz);
if (a == b) {
fb = fa;
} else {
for (int i = 0; i < b.size(); i++) {
fb[i] = C(b[i].val() & ((1 << 15) - 1), b[i].val() >> 15);
}
fft(fb, sz);
}
real ratio = 0.25 / sz;
C r2(0, -1), r3(ratio, 0), r4(0, -ratio), r5(0, 1);
for (int i = 0; i <= (sz >> 1); i++) {
int j = (sz - i) & (sz - 1);
C a1 = (fa[i] + fa[j].conj());
C a2 = (fa[i] - fa[j].conj()) * r2;
C b1 = (fb[i] + fb[j].conj()) * r3;
C b2 = (fb[i] - fb[j].conj()) * r4;
if (i != j) {
C c1 = (fa[j] + fa[i].conj());
C c2 = (fa[j] - fa[i].conj()) * r2;
C d1 = (fb[j] + fb[i].conj()) * r3;
C d2 = (fb[j] - fb[i].conj()) * r4;
fa[i] = c1 * d1 + c2 * d2 * r5;
fb[i] = c1 * d2 + c2 * d1;
}
fa[j] = a1 * b1 + a2 * b2 * r5;
fb[j] = a1 * b2 + a2 * b1;
}
fft(fa, sz);
fft(fb, sz);
vector<T> ret(need);
for (int i = 0; i < need; i++) {
int64_t aa = llround(fa[i].x);
int64_t bb = llround(fb[i].x);
int64_t cc = llround(fa[i].y);
aa = T(aa).val(), bb = T(bb).val(), cc = T(cc).val();
ret[i] = aa + (bb << 15) + (cc << 30);
}
return ret;
}
};
#line 4 "math/fps/formal-power-series.hpp"
/**
* @brief Formal Power Series(形式的冪級数)
*/
template <typename T>
struct FormalPowerSeries : vector<T> {
using vector<T>::vector;
using P = FormalPowerSeries;
using Conv = ArbitraryModConvolution<T>;
P pre(int deg) const {
return P(begin(*this), begin(*this) + min((int)this->size(), deg));
}
P rev(int deg = -1) const {
P ret(*this);
if (deg != -1) ret.resize(deg, T(0));
reverse(begin(ret), end(ret));
return ret;
}
void shrink() {
while (this->size() && this->back() == T(0)) this->pop_back();
}
P operator+(const P &r) const { return P(*this) += r; }
P operator+(const T &v) const { return P(*this) += v; }
P operator-(const P &r) const { return P(*this) -= r; }
P operator-(const T &v) const { return P(*this) -= v; }
P operator*(const P &r) const { return P(*this) *= r; }
P operator*(const T &v) const { return P(*this) *= v; }
P operator/(const P &r) const { return P(*this) /= r; }
P operator%(const P &r) const { return P(*this) %= r; }
P &operator+=(const P &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < r.size(); i++) (*this)[i] += r[i];
return *this;
}
P &operator-=(const P &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < r.size(); i++) (*this)[i] -= r[i];
return *this;
}
// https://judge.yosupo.jp/problem/convolution_mod
P &operator*=(const P &r) {
if (this->empty() || r.empty()) {
this->clear();
return *this;
}
auto ret = Conv::multiply(*this, r);
return *this = {begin(ret), end(ret)};
}
P &operator/=(const P &r) {
if (this->size() < r.size()) {
this->clear();
return *this;
}
int n = this->size() - r.size() + 1;
return *this = (rev().pre(n) * r.rev().inv(n)).pre(n).rev(n);
}
P &operator%=(const P &r) { return *this -= *this / r * r; }
// https://judge.yosupo.jp/problem/division_of_polynomials
pair<P, P> div_mod(const P &r) {
P q = *this / r;
return make_pair(q, *this - q * r);
}
P operator-() const {
P ret(this->size());
for (int i = 0; i < this->size(); i++) ret[i] = -(*this)[i];
return ret;
}
P &operator+=(const T &r) {
if (this->empty()) this->resize(1);
(*this)[0] += r;
return *this;
}
P &operator-=(const T &r) {
if (this->empty()) this->resize(1);
(*this)[0] -= r;
return *this;
}
P &operator*=(const T &v) {
for (int i = 0; i < this->size(); i++) (*this)[i] *= v;
return *this;
}
P dot(P r) const {
P ret(min(this->size(), r.size()));
for (int i = 0; i < ret.size(); i++) ret[i] = (*this)[i] * r[i];
return ret;
}
P operator>>(int sz) const {
if (this->size() <= sz) return {};
P ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
P operator<<(int sz) const {
P ret(*this);
ret.insert(ret.begin(), sz, T(0));
return ret;
}
T operator()(T x) const {
T r = 0, w = 1;
for (auto &v : *this) {
r += w * v;
w *= x;
}
return r;
}
P diff() const {
const int n = (int)this->size();
P ret(max(0, n - 1));
for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i);
return ret;
}
P integral() const {
const int n = (int)this->size();
P ret(n + 1);
ret[0] = T(0);
for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1);
return ret;
}
// https://judge.yosupo.jp/problem/inv_of_formal_power_series
// F(0) must not be 0
P inv(int deg = -1) const {
assert(((*this)[0]) != T(0));
const int n = (int)this->size();
if (deg == -1) deg = n;
P ret({T(1) / (*this)[0]});
for (int i = 1; i < deg; i <<= 1) {
ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1);
}
return ret.pre(deg);
}
// https://judge.yosupo.jp/problem/log_of_formal_power_series
// F(0) must be 1
P log(int deg = -1) const {
assert((*this)[0] == T(1));
const int n = (int)this->size();
if (deg == -1) deg = n;
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
// https://judge.yosupo.jp/problem/sqrt_of_formal_power_series
P sqrt(
int deg = -1,
const function<T(T)> &get_sqrt = [](T) { return T(1); }) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
if ((*this)[0] == T(0)) {
for (int i = 1; i < n; i++) {
if ((*this)[i] != T(0)) {
if (i & 1) return {};
if (deg - i / 2 <= 0) break;
auto ret = (*this >> i).sqrt(deg - i / 2, get_sqrt);
if (ret.empty()) return {};
ret = ret << (i / 2);
if (ret.size() < deg) ret.resize(deg, T(0));
return ret;
}
}
return P(deg, 0);
}
auto sqr = T(get_sqrt((*this)[0]));
if (sqr * sqr != (*this)[0]) return {};
P ret{sqr};
T inv2 = T(1) / T(2);
for (int i = 1; i < deg; i <<= 1) {
ret = (ret + pre(i << 1) * ret.inv(i << 1)) * inv2;
}
return ret.pre(deg);
}
P sqrt(const function<T(T)> &get_sqrt, int deg = -1) const {
return sqrt(deg, get_sqrt);
}
// https://judge.yosupo.jp/problem/exp_of_formal_power_series
// F(0) must be 0
P exp(int deg = -1) const {
if (deg == -1) deg = this->size();
assert((*this)[0] == T(0));
const int n = (int)this->size();
if (deg == -1) deg = n;
P ret({T(1)});
for (int i = 1; i < deg; i <<= 1) {
ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1);
}
return ret.pre(deg);
}
// https://judge.yosupo.jp/problem/pow_of_formal_power_series
P pow(int64_t k, int deg = -1) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
if (k == 0) {
P ret(deg, T(0));
ret[0] = T(1);
return ret;
}
for (int i = 0; i < n; i++) {
if (i * k > deg) return P(deg, T(0));
if ((*this)[i] != T(0)) {
T rev = T(1) / (*this)[i];
P ret = (((*this * rev) >> i).log() * k).exp() * ((*this)[i].pow(k));
ret = (ret << (i * k)).pre(deg);
if (ret.size() < deg) ret.resize(deg, T(0));
return ret;
}
}
return *this;
}
// https://yukicoder.me/problems/no/215
P mod_pow(int64_t k, P g) const {
P modinv = g.rev().inv();
auto get_div = [&](P base) {
if (base.size() < g.size()) {
base.clear();
return base;
}
int n = base.size() - g.size() + 1;
return (base.rev().pre(n) * modinv.pre(n)).pre(n).rev(n);
};
P x(*this), ret{1};
while (k > 0) {
if (k & 1) {
ret *= x;
ret -= get_div(ret) * g;
ret.shrink();
}
x *= x;
x -= get_div(x) * g;
x.shrink();
k >>= 1;
}
return ret;
}
// https://judge.yosupo.jp/problem/polynomial_taylor_shift
P taylor_shift(T c) const {
int n = (int)this->size();
vector<T> fact(n), rfact(n);
fact[0] = rfact[0] = T(1);
for (int i = 1; i < n; i++) fact[i] = fact[i - 1] * T(i);
rfact[n - 1] = T(1) / fact[n - 1];
for (int i = n - 1; i > 1; i--) rfact[i - 1] = rfact[i] * T(i);
P p(*this);
for (int i = 0; i < n; i++) p[i] *= fact[i];
p = p.rev();
P bs(n, T(1));
for (int i = 1; i < n; i++) bs[i] = bs[i - 1] * c * rfact[i] * fact[i - 1];
p = (p * bs).pre(n);
p = p.rev();
for (int i = 0; i < n; i++) p[i] *= rfact[i];
return p;
}
};
template <typename Mint>
using FPS = FormalPowerSeries<Mint>;