Luzhiled's Library
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test/unittest/enumerate-coefficients-of-powers.test.cpp
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- Last update: 2026-08-02 17:14:47+00:00
Depends on
- Montgomery Mod Int (math/combinatorics/montgomery-mod-int.hpp)
- Number Theoretic Transform Friendly ModInt (math/fft/number-theoretic-transform-friendly-mod-int.hpp)
- Enumerate Coefficients of Powers of Formal Power Series (math/fps/enumerate-coefficients-of-powers.hpp)
- Formal Power Series Friendly NTT (NTTmod用形式的冪級数) (math/fps/formal-power-series-friendly-ntt.hpp)
- template/template.hpp
Code
// competitive-verifier: STANDALONE
#include "../../template/template.hpp"
#include "../../math/combinatorics/montgomery-mod-int.hpp"
#include "../../math/fps/enumerate-coefficients-of-powers.hpp"
using mint = modint998244353;
using Poly = FormalPowerSeriesFriendlyNTT<mint>;
Poly multiply_naive(const Poly &a, const Poly &b, int deg) {
Poly result(deg);
for (int i = 0; i < (int)a.size() && i < deg; i++) {
for (int j = 0; j < (int)b.size() && i + j < deg; j++) {
result[i + j] += a[i] * b[j];
}
}
return result;
}
Poly enumerate_coefficients_of_powers_naive(const Poly &f, const Poly &g,
int m) {
const int n = (int)f.size() - 1;
Poly result(m + 1), power(n + 1);
power[0] = mint(1);
for (int i = 0; i <= m; i++) {
result[i] = multiply_naive(power, g, n + 1)[n];
power = multiply_naive(power, f, n + 1);
}
return result;
}
int main() {
mt19937_64 rng(123456789);
for (int n = 0; n <= 20; n++) {
for (int rep = 0; rep < 20; rep++) {
for (int nonzero_constant = 0; nonzero_constant < 2;
nonzero_constant++) {
Poly f(n + 1), g(n + 1);
if (nonzero_constant) {
do {
f[0] = mint((int64_t)rng());
} while (f[0] == mint(0));
}
for (int i = 1; i <= n; i++) f[i] = mint((int64_t)rng());
for (auto &value : g) value = mint((int64_t)rng());
const int m = rng() % (2 * n + 6);
assert(enumerate_coefficients_of_powers<mint>(f, g, m) ==
enumerate_coefficients_of_powers_naive(f, g, m));
}
}
}
}
#line 1 "test/unittest/enumerate-coefficients-of-powers.test.cpp"
// competitive-verifier: STANDALONE
#line 1 "template/template.hpp"
#include <bits/stdc++.h>
#if __has_include(<atcoder/all>)
#include <atcoder/all>
#endif
using namespace std;
using int64 = long long;
const int64 infll = (1LL << 62) - 1;
const int inf = (1 << 30) - 1;
struct IoSetup {
IoSetup() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(10);
cerr << fixed << setprecision(10);
}
} iosetup;
template <typename T1, typename T2>
ostream& operator<<(ostream& os, const pair<T1, T2>& p) {
os << p.first << " " << p.second;
return os;
}
template <typename T1, typename T2>
istream& operator>>(istream& is, pair<T1, T2>& p) {
is >> p.first >> p.second;
return is;
}
template <typename T>
ostream& operator<<(ostream& os, const vector<T>& v) {
for (size_t i = 0; i < v.size(); i++) {
os << v[i] << (i + 1 != v.size() ? " " : "");
}
return os;
}
template <typename T>
istream& operator>>(istream& is, vector<T>& v) {
for (T& in : v) is >> in;
return is;
}
template <typename T1, typename T2>
bool chmax(T1& a, T2 b) {
return a < b && (a = b, true);
}
template <typename T1, typename T2>
bool chmin(T1& a, T2 b) {
return a > b && (a = b, true);
}
template <typename T = int64>
vector<T> make_v(size_t a) {
return vector<T>(a);
}
template <typename T, typename... Ts>
auto make_v(size_t a, Ts... ts) {
return vector<decltype(make_v<T>(ts...))>(a, make_v<T>(ts...));
}
template <typename T, typename V>
enable_if_t<is_class_v<T> == 0> fill_v(T& t, const V& v) {
t = v;
}
template <typename T, typename V>
enable_if_t<is_class_v<T> != 0> fill_v(T& t, const V& v) {
for (auto& e : t) fill_v(e, v);
}
template <typename F>
struct FixPoint : F {
explicit FixPoint(F&& f) : F(std::forward<F>(f)) {}
template <typename... Args>
decltype(auto) operator()(Args&&... args) const {
return F::operator()(*this, std::forward<Args>(args)...);
}
};
template <typename F>
decltype(auto) MFP(F&& f) {
return FixPoint<F>{std::forward<F>(f)};
}
#line 4 "test/unittest/enumerate-coefficients-of-powers.test.cpp"
#line 2 "math/combinatorics/montgomery-mod-int.hpp"
template <uint32_t mod_, bool fast = false>
struct MontgomeryModInt {
private:
using mint = MontgomeryModInt;
using i32 = int32_t;
using i64 = int64_t;
using u32 = uint32_t;
using u64 = uint64_t;
static constexpr u32 get_r() {
u32 ret = mod_;
for (i32 i = 0; i < 4; i++) ret *= 2 - mod_ * ret;
return ret;
}
static constexpr u32 r = get_r();
static constexpr u32 n2 = -u64(mod_) % mod_;
static_assert(r * mod_ == 1, "invalid, r * mod != 1");
static_assert(mod_ < (1 << 30), "invalid, mod >= 2 ^ 30");
static_assert((mod_ & 1) == 1, "invalid, mod % 2 == 0");
u32 x;
public:
MontgomeryModInt() : x{} {}
MontgomeryModInt(const i64& a)
: x(reduce(u64(fast ? a : (a % mod() + mod())) * n2)) {}
static constexpr u32 reduce(const u64& b) {
return u32(b >> 32) + mod() - u32((u64(u32(b) * r) * mod()) >> 32);
}
mint& operator+=(const mint& p) {
if (i32(x += p.x - 2 * mod()) < 0) x += 2 * mod();
return *this;
}
mint& operator-=(const mint& p) {
if (i32(x -= p.x) < 0) x += 2 * mod();
return *this;
}
mint& operator*=(const mint& p) {
x = reduce(u64(x) * p.x);
return *this;
}
mint& operator/=(const mint& p) {
*this *= p.inv();
return *this;
}
mint operator-() const { return mint() - *this; }
mint operator+(const mint& p) const { return mint(*this) += p; }
mint operator-(const mint& p) const { return mint(*this) -= p; }
mint operator*(const mint& p) const { return mint(*this) *= p; }
mint operator/(const mint& p) const { return mint(*this) /= p; }
bool operator==(const mint& p) const {
return (x >= mod() ? x - mod() : x) == (p.x >= mod() ? p.x - mod() : p.x);
}
bool operator!=(const mint& p) const {
return (x >= mod() ? x - mod() : x) != (p.x >= mod() ? p.x - mod() : p.x);
}
u32 val() const {
u32 ret = reduce(x);
return ret >= mod() ? ret - mod() : ret;
}
mint pow(u64 n) const {
mint ret(1), mul(*this);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
mint inv() const { return pow(mod() - 2); }
friend ostream& operator<<(ostream& os, const mint& p) {
return os << p.val();
}
friend istream& operator>>(istream& is, mint& a) {
i64 t;
is >> t;
a = mint(t);
return is;
}
static constexpr u32 mod() { return mod_; }
};
template <uint32_t mod>
using modint = MontgomeryModInt<mod>;
using modint998244353 = modint<998244353>;
using modint1000000007 = modint<1000000007>;
#line 2 "math/fps/enumerate-coefficients-of-powers.hpp"
#line 2 "math/fps/formal-power-series-friendly-ntt.hpp"
#line 1 "math/fft/number-theoretic-transform-friendly-mod-int.hpp"
/**
* @brief Number Theoretic Transform Friendly ModInt
*/
template <typename Mint>
struct NumberTheoreticTransformFriendlyModInt {
static vector<Mint> roots, iroots, rate2, irate2, rate3, irate3;
static int max_base;
NumberTheoreticTransformFriendlyModInt() = default;
static void init() {
if (roots.empty()) {
const unsigned mod = Mint::mod();
assert(mod >= 3 && mod % 2 == 1);
auto tmp = mod - 1;
max_base = 0;
while (tmp % 2 == 0) tmp >>= 1, max_base++;
Mint root = 2;
while (root.pow((mod - 1) >> 1) == 1) {
root += 1;
}
assert(root.pow(mod - 1) == 1);
roots.resize(max_base + 1);
iroots.resize(max_base + 1);
rate2.resize(max_base + 1);
irate2.resize(max_base + 1);
rate3.resize(max_base + 1);
irate3.resize(max_base + 1);
roots[max_base] = root.pow((mod - 1) >> max_base);
iroots[max_base] = Mint(1) / roots[max_base];
for (int i = max_base - 1; i >= 0; i--) {
roots[i] = roots[i + 1] * roots[i + 1];
iroots[i] = iroots[i + 1] * iroots[i + 1];
}
{
Mint prod = 1, iprod = 1;
for (int i = 0; i <= max_base - 2; i++) {
rate2[i] = roots[i + 2] * prod;
irate2[i] = iroots[i + 2] * iprod;
prod *= iroots[i + 2];
iprod *= roots[i + 2];
}
}
{
Mint prod = 1, iprod = 1;
for (int i = 0; i <= max_base - 3; i++) {
rate3[i] = roots[i + 3] * prod;
irate3[i] = iroots[i + 3] * iprod;
prod *= iroots[i + 3];
iprod *= roots[i + 3];
}
}
}
}
static void ntt(vector<Mint>& a) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
int h = __builtin_ctz(n);
assert(h <= max_base);
int len = 0;
Mint imag = roots[2];
if (h & 1) {
int p = 1 << (h - 1);
for (int i = 0; i < p; i++) {
auto r = a[i + p];
a[i + p] = a[i] - r;
a[i] += r;
}
len++;
}
for (; len + 1 < h; len += 2) {
int p = 1 << (h - len - 2);
{ // s = 0
for (int i = 0; i < p; i++) {
auto a0 = a[i];
auto a1 = a[i + p];
auto a2 = a[i + 2 * p];
auto a3 = a[i + 3 * p];
auto a1na3imag = (a1 - a3) * imag;
auto a0a2 = a0 + a2;
auto a1a3 = a1 + a3;
auto a0na2 = a0 - a2;
a[i] = a0a2 + a1a3;
a[i + 1 * p] = a0a2 - a1a3;
a[i + 2 * p] = a0na2 + a1na3imag;
a[i + 3 * p] = a0na2 - a1na3imag;
}
}
Mint rot = rate3[0];
for (int s = 1; s < (1 << len); s++) {
int offset = s << (h - len);
Mint rot2 = rot * rot;
Mint rot3 = rot2 * rot;
for (int i = 0; i < p; i++) {
auto a0 = a[i + offset];
auto a1 = a[i + offset + p] * rot;
auto a2 = a[i + offset + 2 * p] * rot2;
auto a3 = a[i + offset + 3 * p] * rot3;
auto a1na3imag = (a1 - a3) * imag;
auto a0a2 = a0 + a2;
auto a1a3 = a1 + a3;
auto a0na2 = a0 - a2;
a[i + offset] = a0a2 + a1a3;
a[i + offset + 1 * p] = a0a2 - a1a3;
a[i + offset + 2 * p] = a0na2 + a1na3imag;
a[i + offset + 3 * p] = a0na2 - a1na3imag;
}
rot *= rate3[__builtin_ctz(~s)];
}
}
}
static void intt(vector<Mint>& a, bool f = true) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
int h = __builtin_ctz(n);
assert(h <= max_base);
int len = h;
Mint iimag = iroots[2];
for (; len > 1; len -= 2) {
int p = 1 << (h - len);
{ // s = 0
for (int i = 0; i < p; i++) {
auto a0 = a[i];
auto a1 = a[i + 1 * p];
auto a2 = a[i + 2 * p];
auto a3 = a[i + 3 * p];
auto a2na3iimag = (a2 - a3) * iimag;
auto a0na1 = a0 - a1;
auto a0a1 = a0 + a1;
auto a2a3 = a2 + a3;
a[i] = a0a1 + a2a3;
a[i + 1 * p] = (a0na1 + a2na3iimag);
a[i + 2 * p] = (a0a1 - a2a3);
a[i + 3 * p] = (a0na1 - a2na3iimag);
}
}
Mint irot = irate3[0];
for (int s = 1; s < (1 << (len - 2)); s++) {
int offset = s << (h - len + 2);
Mint irot2 = irot * irot;
Mint irot3 = irot2 * irot;
for (int i = 0; i < p; i++) {
auto a0 = a[i + offset];
auto a1 = a[i + offset + 1 * p];
auto a2 = a[i + offset + 2 * p];
auto a3 = a[i + offset + 3 * p];
auto a2na3iimag = (a2 - a3) * iimag;
auto a0na1 = a0 - a1;
auto a0a1 = a0 + a1;
auto a2a3 = a2 + a3;
a[i + offset] = a0a1 + a2a3;
a[i + offset + 1 * p] = (a0na1 + a2na3iimag) * irot;
a[i + offset + 2 * p] = (a0a1 - a2a3) * irot2;
a[i + offset + 3 * p] = (a0na1 - a2na3iimag) * irot3;
}
irot *= irate3[__builtin_ctz(~s)];
}
}
if (len >= 1) {
int p = 1 << (h - 1);
for (int i = 0; i < p; i++) {
auto ajp = a[i] - a[i + p];
a[i] += a[i + p];
a[i + p] = ajp;
}
}
if (f) {
Mint inv_sz = Mint(1) / n;
for (int i = 0; i < n; i++) a[i] *= inv_sz;
}
}
/**
* @brief Transpose of ntt()
*/
static void transposed_ntt(vector<Mint>& a) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
const int h = __builtin_ctz(n);
assert(h <= max_base);
int len = h;
const Mint imag = roots[2];
while (len > 0) {
if (len == 1) {
const int p = 1 << (h - len);
Mint rot = 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 lhs = a[i + offset];
const auto rhs = a[i + offset + p];
a[i + offset] = lhs + rhs;
a[i + offset + p] = (lhs - rhs) * rot;
}
rot *= rate2[__builtin_ctz(~s)];
}
len--;
} else {
const int p = 1 << (h - len);
Mint rot = 1;
for (int s = 0; s < (1 << (len - 2)); s++) {
const int offset = s << (h - len + 2);
const Mint rot2 = rot * rot;
const Mint rot3 = rot2 * rot;
for (int i = 0; i < p; i++) {
const auto a0 = a[i + offset];
const auto a1 = a[i + offset + p];
const auto a2 = a[i + offset + 2 * p];
const auto a3 = a[i + offset + 3 * p];
const auto x = (a2 - a3) * imag;
a[i + offset] = a0 + a1 + a2 + a3;
a[i + offset + p] = (a0 - a1 + x) * rot;
a[i + offset + 2 * p] = (a0 + a1 - a2 - a3) * rot2;
a[i + offset + 3 * p] = (a0 - a1 - x) * rot3;
}
rot *= rate3[__builtin_ctz(~s)];
}
len -= 2;
}
}
}
/**
* @brief Transpose of intt()
*/
static void transposed_intt(vector<Mint>& a, bool f = true) {
init();
const int n = (int)a.size();
assert((n & (n - 1)) == 0);
const int h = __builtin_ctz(n);
assert(h <= max_base);
if (f) {
const Mint inv_sz = Mint(1) / n;
for (auto& value : a) value *= inv_sz;
}
int len = 0;
const Mint iimag = iroots[2];
while (len < h) {
if (len == h - 1) {
const int p = 1 << (h - len - 1);
Mint irot = 1;
for (int s = 0; s < (1 << len); s++) {
const int offset = s << (h - len);
for (int i = 0; i < p; i++) {
const auto lhs = a[i + offset];
const auto rhs = a[i + offset + p] * irot;
a[i + offset] = lhs + rhs;
a[i + offset + p] = lhs - rhs;
}
irot *= irate2[__builtin_ctz(~s)];
}
len++;
} else {
const int p = 1 << (h - len - 2);
Mint irot = 1;
for (int s = 0; s < (1 << len); s++) {
const Mint irot2 = irot * irot;
const Mint irot3 = irot2 * irot;
const int offset = s << (h - len);
for (int i = 0; i < p; i++) {
const auto a0 = a[i + offset];
const auto a1 = a[i + offset + p] * irot;
const auto a2 = a[i + offset + 2 * p] * irot2;
const auto a3 = a[i + offset + 3 * p] * irot3;
const auto x = (a1 - a3) * iimag;
a[i + offset] = a0 + a2 + a1 + a3;
a[i + offset + p] = a0 + a2 - a1 - a3;
a[i + offset + 2 * p] = a0 - a2 + x;
a[i + offset + 3 * p] = a0 - a2 - x;
}
irot *= irate3[__builtin_ctz(~s)];
}
len += 2;
}
}
}
static vector<Mint> multiply(vector<Mint> a, vector<Mint> b) {
int need = a.size() + b.size() - 1;
int nbase = 1;
while ((1 << nbase) < need) nbase++;
int sz = 1 << nbase;
a.resize(sz, 0);
b.resize(sz, 0);
ntt(a);
ntt(b);
Mint inv_sz = Mint(1) / sz;
for (int i = 0; i < sz; i++) a[i] *= b[i] * inv_sz;
intt(a, false);
a.resize(need);
return a;
}
};
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::roots =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::iroots =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::rate2 =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::irate2 =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::rate3 =
vector<Mint>();
template <typename Mint>
vector<Mint> NumberTheoreticTransformFriendlyModInt<Mint>::irate3 =
vector<Mint>();
template <typename Mint>
int NumberTheoreticTransformFriendlyModInt<Mint>::max_base = 0;
#line 4 "math/fps/formal-power-series-friendly-ntt.hpp"
template <typename T>
struct FormalPowerSeriesFriendlyNTT : vector<T> {
using vector<T>::vector;
using P = FormalPowerSeriesFriendlyNTT;
using NTT = NumberTheoreticTransformFriendlyModInt<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 < (int)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 < (int)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 = NTT::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) {
*this -= *this / r * r;
shrink();
return *this;
}
// https://judge.yosupo.jp/problem/division_of_polynomials
pair<P, P> div_mod(const P& r) {
P q = *this / r;
P x = *this - q * r;
x.shrink();
return make_pair(q, x);
}
P operator-() const {
P ret(this->size());
for (int i = 0; i < (int)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 < (int)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 < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
return ret;
}
P operator>>(int sz) const {
if ((int)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 res(deg);
res[0] = {T(1) / (*this)[0]};
for (int d = 1; d < deg; d <<= 1) {
P f(2 * d), g(2 * d);
for (int j = 0; j < min(n, 2 * d); j++) f[j] = (*this)[j];
for (int j = 0; j < d; j++) g[j] = res[j];
NTT::ntt(f);
NTT::ntt(g);
f = f.dot(g);
NTT::intt(f);
for (int j = 0; j < d; j++) f[j] = 0;
NTT::ntt(f);
for (int j = 0; j < 2 * d; j++) f[j] *= g[j];
NTT::intt(f);
for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j];
}
return res;
}
// 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 ((int)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));
P inv;
inv.reserve(deg + 1);
inv.push_back(T(0));
inv.push_back(T(1));
auto inplace_integral = [&](P& F) -> void {
const int n = (int)F.size();
auto mod = T::mod();
while ((int)inv.size() <= n) {
int i = inv.size();
inv.push_back((-inv[mod % i]) * (mod / i));
}
F.insert(begin(F), T(0));
for (int i = 1; i <= n; i++) F[i] *= inv[i];
};
auto inplace_diff = [](P& F) -> void {
if (F.empty()) return;
F.erase(begin(F));
T coeff = 1, one = 1;
for (int i = 0; i < (int)F.size(); i++) {
F[i] *= coeff;
coeff += one;
}
};
P 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);
NTT::ntt(y);
z1 = z2;
P z(m);
for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
NTT::intt(z);
fill(begin(z), begin(z) + m / 2, T(0));
NTT::ntt(z);
for (int i = 0; i < m; ++i) z[i] *= -z1[i];
NTT::intt(z);
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c;
z2.resize(2 * m);
NTT::ntt(z2);
P x(begin(*this), begin(*this) + min<int>(this->size(), m));
inplace_diff(x);
x.push_back(T(0));
NTT::ntt(x);
for (int i = 0; i < m; ++i) x[i] *= y[i];
NTT::intt(x);
x -= b.diff();
x.resize(2 * m);
for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = T(0);
NTT::ntt(x);
for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
NTT::intt(x);
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, T(0));
NTT::ntt(x);
for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
NTT::intt(x);
b.insert(end(b), begin(x) + m, end(x));
}
return P{begin(b), begin(b) + 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 ((int)ret.size() < deg) ret.resize(deg, T(0));
return ret;
}
}
return *this;
}
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 = FormalPowerSeriesFriendlyNTT<Mint>;
#line 4 "math/fps/enumerate-coefficients-of-powers.hpp"
/**
* @brief Enumerate Coefficients of Powers of Formal Power Series
*
* Returns [x^n] f(x)^k g(x) for k = 0, 1, ..., m, where n = f.size() - 1.
*
* @note This implementation requires an NTT-friendly modulus.
* @note The implementation is based on the fast power-projection algorithm
* used for FPS composition/compositional inverse.
* @complexity O(n log^2 n + m log m)
*/
template <typename Mint>
FormalPowerSeriesFriendlyNTT<Mint> enumerate_coefficients_of_powers(
FormalPowerSeriesFriendlyNTT<Mint> f,
FormalPowerSeriesFriendlyNTT<Mint> g = {Mint(1)}, int m = -1) {
using Poly = FormalPowerSeriesFriendlyNTT<Mint>;
using NTT = NumberTheoreticTransformFriendlyModInt<Mint>;
assert(!f.empty());
int n = (int)f.size() - 1;
int k = 1;
g.resize(n + 1);
if (m == -1) m = n;
assert(m >= 0);
int h = 1;
while (h < n + 1) h <<= 1;
NTT::init();
// The largest transform has length 2 * h.
assert(1 + __builtin_ctz(h) <= NTT::max_base);
Poly P((n + 1) * k), Q((n + 1) * k), next_P, next_Q;
Poly buffer, doubled;
for (int i = 0; i <= n; i++) P[i * k] = g[i];
for (int i = 0; i <= n; i++) Q[i * k] = -f[i];
Q[0] += Mint(1);
const Mint inv_two = Mint(1) / Mint(2);
const Mint two = Mint(2);
Mint inv_k = Mint(1);
Mint inv_h = Mint(1) / Mint(h);
while (n > 0) {
// If a contains the length-k NTT of A(x), append the evaluations at the
// odd 2k-th roots and turn it into the length-2k NTT of A(x).
auto ntt_doubling = [&](Poly& a) {
assert((int)a.size() == k);
doubled = a;
NTT::intt(doubled, false);
const Mint zeta = NTT::roots[__builtin_ctz(2 * k)];
Mint power = inv_k;
for (int i = 0; i < k; i++) {
doubled[i] *= power;
power *= zeta;
}
NTT::ntt(doubled);
a.insert(a.end(), doubled.begin(), doubled.end());
};
next_P.clear();
next_Q.clear();
next_P.reserve((n + 1) * 2 * k);
next_Q.reserve((n + 1) * 2 * k);
for (int i = 0; i <= n; i++) {
buffer.assign(P.begin() + i * k, P.begin() + (i + 1) * k);
ntt_doubling(buffer);
next_P.insert(next_P.end(), buffer.begin(), buffer.end());
buffer.assign(Q.begin() + i * k, Q.begin() + (i + 1) * k);
if (i == 0) {
// Q has a distinguished constant 1. Remove it before doubling and
// restore it in the two halves afterwards.
for (int j = 0; j < k; j++) buffer[j] -= Mint(1);
ntt_doubling(buffer);
for (int j = 0; j < k; j++) buffer[j] += Mint(1);
for (int j = 0; j < k; j++) buffer[k + j] -= Mint(1);
} else {
ntt_doubling(buffer);
}
next_Q.insert(next_Q.end(), buffer.begin(), buffer.end());
}
// Pad the x direction to length 2h for every frequency in the other
// variable. h * k stays constant, so the total working size is linear.
next_P.resize(4 * h * k);
next_Q.resize(4 * h * k);
Poly p(2 * h), q(2 * h);
const Mint inverse_zeta = NTT::iroots[__builtin_ctz(2 * h)];
vector<int> bit_reversal;
if (n & 1) {
bit_reversal.resize(h);
const int log_h = __builtin_ctz(h);
for (int i = 0; i < h; i++) {
bit_reversal[i] =
(bit_reversal[i >> 1] >> 1) + ((i & 1) << (log_h - 1));
}
}
for (int frequency = 0; frequency < 2 * k; frequency++) {
p.assign(2 * h, Mint(0));
q.assign(2 * h, Mint(0));
for (int i = 0; i < h; i++) {
p[i] = next_P[i * 2 * k + frequency];
q[i] = next_Q[i * 2 * k + frequency];
}
NTT::ntt(p);
NTT::ntt(q);
// In this radix-4 DIF ordering, changing x to -x swaps adjacent NTT
// values.
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[2 * i] * q[2 * i + 1];
if ((n & 1) == 0) {
for (int i = 0; i < h; i++) {
p[i] = (p[2 * i] + p[2 * i + 1]) * inv_two;
}
} else {
Mint coefficient = inv_two;
buffer.resize(h);
for (int i : bit_reversal) {
buffer[i] = (p[2 * i] - p[2 * i + 1]) * coefficient;
coefficient *= inverse_zeta;
}
p.swap(buffer);
}
p.resize(h);
q.resize(h);
NTT::intt(p, false);
NTT::intt(q, false);
for (int i = 0; i < h; i++) {
next_P[i * 2 * k + frequency] = p[i] * inv_h;
next_Q[i * 2 * k + frequency] = q[i] * inv_h;
}
}
next_P.resize((n / 2 + 1) * 2 * k);
next_Q.resize((n / 2 + 1) * 2 * k);
P.swap(next_P);
Q.swap(next_Q);
n /= 2;
h /= 2;
k *= 2;
inv_h *= two;
inv_k *= inv_two;
}
Poly numerator(P.begin(), P.begin() + k);
NTT::intt(numerator, false);
for (auto& value : numerator) value *= inv_k;
if (f[0] == Mint(0)) {
auto result = numerator.rev().pre(m + 1);
result.resize(m + 1);
return result;
}
Poly denominator(Q.begin(), Q.begin() + k);
NTT::intt(denominator, false);
for (auto& value : denominator) value *= inv_k;
denominator[0] -= Mint(1);
Poly shifted_one(k + 1);
shifted_one[k] = Mint(1);
auto reversed_numerator = numerator.rev().pre(m + 1);
auto reversed_denominator = (denominator + shifted_one).rev().pre(m + 1);
auto result =
(reversed_numerator * reversed_denominator.inv(m + 1)).pre(m + 1);
result.resize(m + 1);
return result;
}
#line 7 "test/unittest/enumerate-coefficients-of-powers.test.cpp"
using mint = modint998244353;
using Poly = FormalPowerSeriesFriendlyNTT<mint>;
Poly multiply_naive(const Poly &a, const Poly &b, int deg) {
Poly result(deg);
for (int i = 0; i < (int)a.size() && i < deg; i++) {
for (int j = 0; j < (int)b.size() && i + j < deg; j++) {
result[i + j] += a[i] * b[j];
}
}
return result;
}
Poly enumerate_coefficients_of_powers_naive(const Poly &f, const Poly &g,
int m) {
const int n = (int)f.size() - 1;
Poly result(m + 1), power(n + 1);
power[0] = mint(1);
for (int i = 0; i <= m; i++) {
result[i] = multiply_naive(power, g, n + 1)[n];
power = multiply_naive(power, f, n + 1);
}
return result;
}
int main() {
mt19937_64 rng(123456789);
for (int n = 0; n <= 20; n++) {
for (int rep = 0; rep < 20; rep++) {
for (int nonzero_constant = 0; nonzero_constant < 2;
nonzero_constant++) {
Poly f(n + 1), g(n + 1);
if (nonzero_constant) {
do {
f[0] = mint((int64_t)rng());
} while (f[0] == mint(0));
}
for (int i = 1; i <= n; i++) f[i] = mint((int64_t)rng());
for (auto &value : g) value = mint((int64_t)rng());
const int m = rng() % (2 * n + 6);
assert(enumerate_coefficients_of_powers<mint>(f, g, m) ==
enumerate_coefficients_of_powers_naive(f, g, m));
}
}
}
}