CP-Algorithms Library
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Bell Number (verify/poly/bell.test.cpp)
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- Last update: 2024-11-21 04:16:11+01:00
- Problem: https://judge.yosupo.jp/problem/bell_number
Depends on
- cp-algo/math/affine.hpp
- cp-algo/math/combinatorics.hpp
- cp-algo/math/common.hpp
- cp-algo/math/fft.hpp
- cp-algo/math/poly.hpp
- cp-algo/math/poly/impl/base.hpp
- cp-algo/math/poly/impl/div.hpp
- cp-algo/math/poly/impl/euclid.hpp
- cp-algo/number_theory/discrete_sqrt.hpp
- cp-algo/number_theory/modint.hpp
- cp-algo/random/rng.hpp
- cp-algo/util/complex.hpp
Code
// @brief Bell Number
#define PROBLEM "https://judge.yosupo.jp/problem/bell_number"
#pragma GCC optimize("Ofast,unroll-loops")
#include "cp-algo/math/poly.hpp"
#include <bits/stdc++.h>
using namespace std;
using namespace cp_algo::math;
const int mod = 998244353;
using base = modint<mod>;
using polyn = poly_t<base>;
void solve() {
int n;
cin >> n;
(polyn::expx(n+1) - polyn(1)).exp_inplace(n+1).invborel().print(n+1);
}
signed main() {
//freopen("input.txt", "r", stdin);
ios::sync_with_stdio(0);
cin.tie(0);
int t = 1;
while(t--) {
solve();
}
}
#line 1 "verify/poly/bell.test.cpp"
// @brief Bell Number
#define PROBLEM "https://judge.yosupo.jp/problem/bell_number"
#pragma GCC optimize("Ofast,unroll-loops")
#line 1 "cp-algo/math/poly.hpp"
#line 1 "cp-algo/math/poly/impl/euclid.hpp"
#line 1 "cp-algo/math/affine.hpp"
#include <optional>
#include <utility>
#include <cassert>
#include <tuple>
namespace cp_algo::math {
// a * x + b
template<typename base>
struct lin {
base a = 1, b = 0;
std::optional<base> c;
lin() {}
lin(base b): a(0), b(b) {}
lin(base a, base b): a(a), b(b) {}
lin(base a, base b, base _c): a(a), b(b), c(_c) {}
// polynomial product modulo x^2 - c
lin operator * (const lin& t) {
assert(c && t.c && *c == *t.c);
return {a * t.b + b * t.a, b * t.b + a * t.a * (*c), *c};
}
// a * (t.a * x + t.b) + b
lin apply(lin const& t) const {
return {a * t.a, a * t.b + b};
}
void prepend(lin const& t) {
*this = t.apply(*this);
}
base eval(base x) const {
return a * x + b;
}
};
// (ax+b) / (cx+d)
template<typename base>
struct linfrac {
base a, b, c, d;
linfrac(): a(1), b(0), c(0), d(1) {} // x, identity for composition
linfrac(base a): a(a), b(1), c(1), d(0) {} // a + 1/x, for continued fractions
linfrac(base a, base b, base c, base d): a(a), b(b), c(c), d(d) {}
// composition of two linfracs
linfrac operator * (linfrac t) const {
return t.prepend(linfrac(*this));
}
linfrac operator-() const {
return {-a, -b, -c, -d};
}
linfrac adj() const {
return {d, -b, -c, a};
}
linfrac& prepend(linfrac const& t) {
t.apply(a, c);
t.apply(b, d);
return *this;
}
// apply linfrac to A/B
void apply(base &A, base &B) const {
std::tie(A, B) = std::pair{a * A + b * B, c * A + d * B};
}
};
}
#line 1 "cp-algo/math/fft.hpp"
#line 1 "cp-algo/math/common.hpp"
#include <functional>
#include <cstdint>
namespace cp_algo::math {
#ifdef CP_ALGO_MAXN
const int maxn = CP_ALGO_MAXN;
#else
const int maxn = 1 << 19;
#endif
const int magic = 64; // threshold for sizes to run the naive algo
auto bpow(auto const& x, auto n, auto const& one, auto op) {
if(n == 0) {
return one;
} else {
auto t = bpow(x, n / 2, one, op);
t = op(t, t);
if(n % 2) {
t = op(t, x);
}
return t;
}
}
auto bpow(auto x, auto n, auto ans) {
return bpow(x, n, ans, std::multiplies{});
}
template<typename T>
T bpow(T const& x, auto n) {
return bpow(x, n, T(1));
}
}
#line 1 "cp-algo/number_theory/modint.hpp"
#line 4 "cp-algo/number_theory/modint.hpp"
#include <iostream>
#line 6 "cp-algo/number_theory/modint.hpp"
namespace cp_algo::math {
inline constexpr auto inv2(auto x) {
assert(x % 2);
std::make_unsigned_t<decltype(x)> y = 1;
while(y * x != 1) {
y *= 2 - x * y;
}
return y;
}
template<typename modint, typename _Int>
struct modint_base {
using Int = _Int;
using UInt = std::make_unsigned_t<Int>;
static constexpr size_t bits = sizeof(Int) * 8;
using Int2 = std::conditional_t<bits <= 32, int64_t, __int128_t>;
using UInt2 = std::conditional_t<bits <= 32, uint64_t, __uint128_t>;
static Int mod() {
return modint::mod();
}
static UInt imod() {
return modint::imod();
}
static UInt2 pw128() {
return modint::pw128();
}
static UInt m_reduce(UInt2 ab) {
if(mod() % 2 == 0) [[unlikely]] {
return UInt(ab % mod());
} else {
UInt2 m = (UInt)ab * imod();
return UInt((ab + m * mod()) >> bits);
}
}
static UInt m_transform(UInt a) {
if(mod() % 2 == 0) [[unlikely]] {
return a;
} else {
return m_reduce(a * pw128());
}
}
modint_base(): r(0) {}
modint_base(Int2 rr): r(UInt(rr % mod())) {
r = std::min(r, r + mod());
r = m_transform(r);
}
modint inv() const {
return bpow(to_modint(), mod() - 2);
}
modint operator - () const {
modint neg;
neg.r = std::min(-r, 2 * mod() - r);
return neg;
}
modint& operator /= (const modint &t) {
return to_modint() *= t.inv();
}
modint& operator *= (const modint &t) {
r = m_reduce((UInt2)r * t.r);
return to_modint();
}
modint& operator += (const modint &t) {
r += t.r; r = std::min(r, r - 2 * mod());
return to_modint();
}
modint& operator -= (const modint &t) {
r -= t.r; r = std::min(r, r + 2 * mod());
return to_modint();
}
modint operator + (const modint &t) const {return modint(to_modint()) += t;}
modint operator - (const modint &t) const {return modint(to_modint()) -= t;}
modint operator * (const modint &t) const {return modint(to_modint()) *= t;}
modint operator / (const modint &t) const {return modint(to_modint()) /= t;}
// Why <=> doesn't work?..
auto operator == (const modint_base &t) const {return getr() == t.getr();}
auto operator != (const modint_base &t) const {return getr() != t.getr();}
auto operator <= (const modint_base &t) const {return getr() <= t.getr();}
auto operator >= (const modint_base &t) const {return getr() >= t.getr();}
auto operator < (const modint_base &t) const {return getr() < t.getr();}
auto operator > (const modint_base &t) const {return getr() > t.getr();}
Int rem() const {
UInt R = getr();
return 2 * R > (UInt)mod() ? R - mod() : R;
}
// Only use if you really know what you're doing!
UInt modmod() const {return (UInt)8 * mod() * mod();};
void add_unsafe(UInt t) {r += t;}
void pseudonormalize() {r = std::min(r, r - modmod());}
modint const& normalize() {
if(r >= (UInt)mod()) {
r %= mod();
}
return to_modint();
}
void setr(UInt rr) {r = m_transform(rr);}
UInt getr() const {
UInt res = m_reduce(r);
return std::min(res, res - mod());
}
void setr_direct(UInt rr) {r = rr;}
UInt getr_direct() const {return r;}
private:
UInt r;
modint& to_modint() {return static_cast<modint&>(*this);}
modint const& to_modint() const {return static_cast<modint const&>(*this);}
};
template<typename modint>
concept modint_type = std::is_base_of_v<modint_base<modint, typename modint::Int>, modint>;
template<modint_type modint>
std::istream& operator >> (std::istream &in, modint &x) {
typename modint::UInt r;
auto &res = in >> r;
x.setr(r);
return res;
}
template<modint_type modint>
std::ostream& operator << (std::ostream &out, modint const& x) {
return out << x.getr();
}
template<auto m>
struct modint: modint_base<modint<m>, decltype(m)> {
using Base = modint_base<modint<m>, decltype(m)>;
using Base::Base;
static constexpr Base::UInt im = m % 2 ? inv2(-m) : 0;
static constexpr Base::UInt r2 = (typename Base::UInt2)(-1) % m + 1;
static constexpr Base::Int mod() {return m;}
static constexpr Base::UInt imod() {return im;}
static constexpr Base::UInt2 pw128() {return r2;}
};
template<typename Int = int64_t>
struct dynamic_modint: modint_base<dynamic_modint<Int>, Int> {
using Base = modint_base<dynamic_modint<Int>, Int>;
using Base::Base;
static Int mod() {return m;}
static Base::UInt imod() {return im;}
static Base::UInt2 pw128() {return r2;}
static void switch_mod(Int nm) {
m = nm;
im = m % 2 ? inv2(-m) : 0;
r2 = static_cast<Base::UInt>(static_cast<Base::UInt2>(-1) % m + 1);
}
// Wrapper for temp switching
auto static with_mod(Int tmp, auto callback) {
struct scoped {
Int prev = mod();
~scoped() {switch_mod(prev);}
} _;
switch_mod(tmp);
return callback();
}
private:
static thread_local Int m;
static thread_local Base::UInt im, r2;
};
template<typename Int>
Int thread_local dynamic_modint<Int>::m = 1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::im = -1;
template<typename Int>
dynamic_modint<Int>::Base::UInt thread_local dynamic_modint<Int>::r2 = 0;
}
#line 1 "cp-algo/util/complex.hpp"
#include <cmath>
namespace cp_algo {
template<typename T>
struct complex {
T x, y;
constexpr complex() {}
constexpr complex(T x): x(x), y(0) {}
constexpr complex(T x, T y): x(x), y(y) {}
complex& operator *= (T t) {x *= t; y *= t; return *this;}
complex& operator /= (T t) {x /= t; y /= t; return *this;}
complex operator * (T t) const {return complex(*this) *= t;}
complex operator / (T t) const {return complex(*this) /= t;}
complex& operator += (complex t) {x += t.x; y += t.y; return *this;}
complex& operator -= (complex t) {x -= t.x; y -= t.y; return *this;}
complex operator * (complex t) const {return {x * t.x - y * t.y, x * t.y + y * t.x};}
complex operator / (complex t) const {return *this * t.conj() / t.norm();}
complex operator + (complex t) const {return complex(*this) += t;}
complex operator - (complex t) const {return complex(*this) -= t;}
complex& operator *= (complex t) {return *this = *this * t;}
complex& operator /= (complex t) {return *this = *this / t;}
complex operator - () const {return {-x, -y};}
complex conj() const {return {x, -y};}
T norm() const {return x * x + y * y;}
T abs() const {return std::sqrt(norm());}
T real() const {return x;}
T imag() const {return y;}
static complex polar(T r, T theta) {return {r * std::cos(theta), r * std::sin(theta)};}
auto operator <=> (complex const& t) const = default;
};
template<typename T>
complex<T> operator * (auto x, complex<T> y) {return y * x;}
template<typename T> complex<T> conj(complex<T> x) {return x.conj();}
template<typename T> T norm(complex<T> x) {return x.norm();}
template<typename T> T abs(complex<T> x) {return x.abs();}
template<typename T> T real(complex<T> x) {return x.real();}
template<typename T> T imag(complex<T> x) {return x.imag();}
template<typename T> complex<T> polar(T r, T theta) {return complex<T>::polar(r, theta);}
}
#line 6 "cp-algo/math/fft.hpp"
#include <algorithm>
#include <complex>
#line 9 "cp-algo/math/fft.hpp"
#include <ranges>
#include <vector>
#include <bit>
#include <experimental/simd>
namespace cp_algo::math::fft {
using ftype = double;
using point = complex<ftype>;
using vftype = std::experimental::native_simd<ftype>;
using vpoint = complex<vftype>;
static constexpr size_t flen = vftype::size();
template<typename ft>
constexpr ft to_ft(auto x) {
return ft{} + x;
}
template<typename pt>
constexpr pt to_pt(point r) {
using ft = std::conditional_t<std::is_same_v<point, pt>, ftype, vftype>;
return {to_ft<ft>(r.real()), to_ft<ft>(r.imag())};
}
struct cvector {
static constexpr size_t pre_roots = 1 << 17;
std::vector<vftype> x, y;
cvector(size_t n) {
n = std::max(flen, std::bit_ceil(n));
x.resize(n / flen);
y.resize(n / flen);
}
template<class pt = point>
void set(size_t k, pt t) {
if constexpr(std::is_same_v<pt, point>) {
x[k / flen][k % flen] = real(t);
y[k / flen][k % flen] = imag(t);
} else {
x[k / flen] = real(t);
y[k / flen] = imag(t);
}
}
template<class pt = point>
pt get(size_t k) const {
if constexpr(std::is_same_v<pt, point>) {
return {x[k / flen][k % flen], y[k / flen][k % flen]};
} else {
return {x[k / flen], y[k / flen]};
}
}
vpoint vget(size_t k) const {
return get<vpoint>(k);
}
size_t size() const {
return flen * std::size(x);
}
void dot(cvector const& t) {
size_t n = size();
for(size_t k = 0; k < n; k += flen) {
set(k, get<vpoint>(k) * t.get<vpoint>(k));
}
}
static const cvector roots;
template<class pt = point>
static pt root(size_t n, size_t k) {
if(n < pre_roots) {
return roots.get<pt>(n + k);
} else {
auto arg = std::numbers::pi / (ftype)n;
if constexpr(std::is_same_v<pt, point>) {
return {cos((ftype)k * arg), sin((ftype)k * arg)};
} else {
return pt{vftype{[&](auto i) {return cos(ftype(k + i) * arg);}},
vftype{[&](auto i) {return sin(ftype(k + i) * arg);}}};
}
}
}
template<class pt = point>
static void exec_on_roots(size_t n, size_t m, auto &&callback) {
size_t step = sizeof(pt) / sizeof(point);
pt cur;
pt arg = to_pt<pt>(root<point>(n, step));
for(size_t i = 0; i < m; i += step) {
if(i % 32 == 0 || n < pre_roots) {
cur = root<pt>(n, i);
} else {
cur *= arg;
}
callback(i, cur);
}
}
void ifft() {
size_t n = size();
for(size_t i = 1; i < n; i *= 2) {
for(size_t j = 0; j < n; j += 2 * i) {
auto butterfly = [&]<class pt>(size_t k, pt rt) {
k += j;
auto t = get<pt>(k + i) * conj(rt);
set(k + i, get<pt>(k) - t);
set(k, get<pt>(k) + t);
};
if(2 * i <= flen) {
exec_on_roots(i, i, butterfly);
} else {
exec_on_roots<vpoint>(i, i, butterfly);
}
}
}
for(size_t k = 0; k < n; k += flen) {
set(k, get<vpoint>(k) /= to_pt<vpoint>((ftype)n));
}
}
void fft() {
size_t n = size();
for(size_t i = n / 2; i >= 1; i /= 2) {
for(size_t j = 0; j < n; j += 2 * i) {
auto butterfly = [&]<class pt>(size_t k, pt rt) {
k += j;
auto A = get<pt>(k) + get<pt>(k + i);
auto B = get<pt>(k) - get<pt>(k + i);
set(k, A);
set(k + i, B * rt);
};
if(2 * i <= flen) {
exec_on_roots(i, i, butterfly);
} else {
exec_on_roots<vpoint>(i, i, butterfly);
}
}
}
}
};
const cvector cvector::roots = []() {
cvector res(pre_roots);
for(size_t n = 1; n < res.size(); n *= 2) {
auto base = polar<ftype>(1., std::numbers::pi / (ftype)n);
point cur = 1;
for(size_t k = 0; k < n; k++) {
if((k & 15) == 0) {
cur = polar<ftype>(1., std::numbers::pi * (ftype)k / (ftype)n);
}
res.set(n + k, cur);
cur *= base;
}
}
return res;
}();
template<typename base>
struct dft {
cvector A;
dft(std::vector<base> const& a, size_t n): A(n) {
for(size_t i = 0; i < std::min(n, a.size()); i++) {
A.set(i, a[i]);
}
if(n) {
A.fft();
}
}
std::vector<base> operator *= (dft const& B) {
assert(A.size() == B.A.size());
size_t n = A.size();
if(!n) {
return std::vector<base>();
}
A.dot(B.A);
A.ifft();
std::vector<base> res(n);
for(size_t k = 0; k < n; k++) {
res[k] = A.get(k);
}
return res;
}
auto operator * (dft const& B) const {
return dft(*this) *= B;
}
point operator [](int i) const {return A.get(i);}
};
template<modint_type base>
struct dft<base> {
int split;
cvector A, B;
dft(auto const& a, size_t n): A(n), B(n) {
split = int(std::sqrt(base::mod()));
cvector::exec_on_roots(2 * n, size(a), [&](size_t i, point rt) {
size_t ti = std::min(i, i - n);
A.set(ti, A.get(ti) + ftype(a[i].rem() % split) * rt);
B.set(ti, B.get(ti) + ftype(a[i].rem() / split) * rt);
});
if(n) {
A.fft();
B.fft();
}
}
void mul(auto &&C, auto const& D, auto &res, size_t k) {
assert(A.size() == C.size());
size_t n = A.size();
if(!n) {
res = {};
return;
}
for(size_t i = 0; i < n; i += flen) {
auto tmp = A.vget(i) * D.vget(i) + B.vget(i) * C.vget(i);
A.set(i, A.vget(i) * C.vget(i));
B.set(i, B.vget(i) * D.vget(i));
C.set(i, tmp);
}
A.ifft();
B.ifft();
C.ifft();
auto splitsplit = (base(split) * split).rem();
cvector::exec_on_roots(2 * n, std::min(n, k), [&](size_t i, point rt) {
rt = conj(rt);
auto Ai = A.get(i) * rt;
auto Bi = B.get(i) * rt;
auto Ci = C.get(i) * rt;
int64_t A0 = llround(real(Ai));
int64_t A1 = llround(real(Ci));
int64_t A2 = llround(real(Bi));
res[i] = A0 + A1 * split + A2 * splitsplit;
if(n + i >= k) {
return;
}
int64_t B0 = llround(imag(Ai));
int64_t B1 = llround(imag(Ci));
int64_t B2 = llround(imag(Bi));
res[n + i] = B0 + B1 * split + B2 * splitsplit;
});
}
void mul_inplace(auto &&B, auto& res, size_t k) {
mul(B.A, B.B, res, k);
}
void mul(auto const& B, auto& res, size_t k) {
mul(cvector(B.A), B.B, res, k);
}
std::vector<base> operator *= (dft &B) {
std::vector<base> res(2 * A.size());
mul_inplace(B, res, size(res));
return res;
}
std::vector<base> operator *= (dft const& B) {
std::vector<base> res(2 * A.size());
mul(B, res, size(res));
return res;
}
auto operator * (dft const& B) const {
return dft(*this) *= B;
}
point operator [](int i) const {return A.get(i);}
};
void mul_slow(auto &a, auto const& b, size_t k) {
if(empty(a) || empty(b)) {
a.clear();
} else {
size_t n = std::min(k, size(a));
size_t m = std::min(k, size(b));
a.resize(k);
for(int j = int(k - 1); j >= 0; j--) {
a[j] *= b[0];
for(int i = std::max(j - (int)n, 0) + 1; i < std::min(j + 1, (int)m); i++) {
a[j] += a[j - i] * b[i];
}
}
}
}
size_t com_size(size_t as, size_t bs) {
if(!as || !bs) {
return 0;
}
return std::max(flen, std::bit_ceil(as + bs - 1) / 2);
}
void mul_truncate(auto &a, auto const& b, size_t k) {
using base = std::decay_t<decltype(a[0])>;
if(std::min({k, size(a), size(b)}) < magic) {
mul_slow(a, b, k);
return;
}
auto n = std::max(flen, std::bit_ceil(
std::min(k, size(a)) + std::min(k, size(b)) - 1
) / 2);
a.resize(k);
auto A = dft<base>(a, n);
if(&a == &b) {
A.mul(A, a, k);
} else {
A.mul_inplace(dft<base>(b | std::views::take(k), n), a, k);
}
}
void mul(auto &a, auto const& b) {
if(size(a)) {
mul_truncate(a, b, size(a) + size(b) - 1);
}
}
}
#line 7 "cp-algo/math/poly/impl/euclid.hpp"
#include <numeric>
#line 11 "cp-algo/math/poly/impl/euclid.hpp"
#include <list>
// operations related to gcd and Euclidean algo
namespace cp_algo::math::poly::impl {
template<typename poly>
using gcd_result = std::pair<
std::list<std::decay_t<poly>>,
linfrac<std::decay_t<poly>>>;
template<typename poly>
gcd_result<poly> half_gcd(poly &&A, poly &&B) {
assert(A.deg() >= B.deg());
size_t m = size(A.a) / 2;
if(B.deg() < (int)m) {
return {};
}
auto [ai, R] = A.divmod(B);
std::tie(A, B) = {B, R};
std::list a = {ai};
auto T = -linfrac(ai).adj();
auto advance = [&](size_t k) {
auto [ak, Tk] = half_gcd(A.div_xk(k), B.div_xk(k));
a.splice(end(a), ak);
T.prepend(Tk);
return Tk;
};
advance(m).apply(A, B);
if constexpr (std::is_reference_v<poly>) {
advance(2 * m - A.deg()).apply(A, B);
} else {
advance(2 * m - A.deg());
}
return {std::move(a), std::move(T)};
}
template<typename poly>
gcd_result<poly> full_gcd(poly &&A, poly &&B) {
using poly_t = std::decay_t<poly>;
std::list<poly_t> ak;
std::vector<linfrac<poly_t>> trs;
while(!B.is_zero()) {
auto [a0, R] = A.divmod(B);
ak.push_back(a0);
trs.push_back(-linfrac(a0).adj());
std::tie(A, B) = {B, R};
auto [a, Tr] = half_gcd(A, B);
ak.splice(end(ak), a);
trs.push_back(Tr);
}
return {ak, std::accumulate(rbegin(trs), rend(trs), linfrac<poly_t>{}, std::multiplies{})};
}
// computes product of linfrac on [L, R)
auto convergent(auto L, auto R) {
using poly = decltype(L)::value_type;
if(R == next(L)) {
return linfrac(*L);
} else {
int s = std::transform_reduce(L, R, 0, std::plus{}, std::mem_fn(&poly::deg));
auto M = L;
for(int c = M->deg(); 2 * c <= s; M++) {
c += next(M)->deg();
}
return convergent(L, M) * convergent(M, R);
}
}
template<typename poly>
poly min_rec(poly const& p, size_t d) {
auto R2 = p.mod_xk(d).reversed(d), R1 = poly::xk(d);
if(R2.is_zero()) {
return poly(1);
}
auto [a, Tr] = full_gcd(R1, R2);
a.emplace_back();
auto pref = begin(a);
for(int delta = (int)d - a.front().deg(); delta >= 0; pref++) {
delta -= pref->deg() + next(pref)->deg();
}
return convergent(begin(a), pref).a;
}
template<typename poly>
std::optional<poly> inv_mod(poly p, poly q) {
assert(!q.is_zero());
auto [a, Tr] = full_gcd(q, p);
if(q.deg() != 0) {
return std::nullopt;
}
return Tr.b / q[0];
}
}
#line 1 "cp-algo/math/poly/impl/base.hpp"
#line 6 "cp-algo/math/poly/impl/base.hpp"
// really basic operations, typically taking O(n)
namespace cp_algo::math::poly::impl {
template<typename polyn>
void normalize(polyn& p) {
while(p.deg() >= 0 && p.lead() == typename polyn::base(0)) {
p.a.pop_back();
}
}
auto neg_inplace(auto &&p) {
std::ranges::transform(p.a, begin(p.a), std::negate{});
return p;
}
auto& scale(auto &p, auto x) {
for(auto &it: p.a) {
it *= x;
}
p.normalize();
return p;
}
auto& add(auto &p, auto const& q) {
p.a.resize(std::max(p.a.size(), q.a.size()));
std::ranges::transform(p.a, q.a, begin(p.a), std::plus{});
normalize(p);
return p;
}
auto& sub(auto &p, auto const& q) {
p.a.resize(std::max(p.a.size(), q.a.size()));
std::ranges::transform(p.a, q.a, begin(p.a), std::minus{});
normalize(p);
return p;
}
auto substr(auto const& p, size_t l, size_t k) {
return std::vector(
begin(p.a) + std::min(l, p.a.size()),
begin(p.a) + std::min(l + k, p.a.size())
);
}
auto& reverse(auto &p, size_t n) {
p.a.resize(n);
std::ranges::reverse(p.a);
normalize(p);
return p;
}
}
#line 1 "cp-algo/math/poly/impl/div.hpp"
#line 6 "cp-algo/math/poly/impl/div.hpp"
// operations related to polynomial division
namespace cp_algo::math::poly::impl {
auto divmod_slow(auto const& p, auto const& q) {
auto R = p;
auto D = decltype(p){};
auto q_lead_inv = q.lead().inv();
while(R.deg() >= q.deg()) {
D.a.push_back(R.lead() * q_lead_inv);
if(D.lead() != 0) {
for(size_t i = 1; i <= q.a.size(); i++) {
R.a[R.a.size() - i] -= D.lead() * q.a[q.a.size() - i];
}
}
R.a.pop_back();
}
std::ranges::reverse(D.a);
R.normalize();
return std::array{D, R};
}
template<typename poly>
auto divmod_hint(poly const& p, poly const& q, poly const& qri) {
assert(!q.is_zero());
int d = p.deg() - q.deg();
if(std::min(d, q.deg()) < magic) {
return divmod_slow(p, q);
}
poly D;
if(d >= 0) {
D = (p.reversed().mod_xk(d + 1) * qri.mod_xk(d + 1)).mod_xk(d + 1).reversed(d + 1);
}
return std::array{D, p - D * q};
}
auto divmod(auto const& p, auto const& q) {
assert(!q.is_zero());
int d = p.deg() - q.deg();
if(std::min(d, q.deg()) < magic) {
return divmod_slow(p, q);
}
return divmod_hint(p, q, q.reversed().inv(d + 1));
}
template<typename poly>
poly powmod_hint(poly const& p, int64_t k, poly const& md, poly const& mdri) {
return bpow(p, k, poly(1), [&](auto const& p, auto const& q){
return divmod_hint(p * q, md, mdri)[1];
});
}
template<typename poly>
auto powmod(poly const& p, int64_t k, poly const& md) {
int d = md.deg();
if(p == poly::xk(1) && false) { // does it actually speed anything up?..
if(k < md.deg()) {
return poly::xk(k);
} else {
auto mdr = md.reversed();
return (mdr.inv(k - md.deg() + 1, md.deg()) * mdr).reversed(md.deg());
}
}
if(md == poly::xk(d)) {
return p.pow(k, d);
}
if(md == poly::xk(d) - poly(1)) {
return p.powmod_circular(k, d);
}
return powmod_hint(p, k, md, md.reversed().inv(md.deg() + 1));
}
template<typename poly>
poly& inv_inplace(poly& q, int64_t k, size_t n) {
using poly_t = std::decay_t<poly>;
using base = poly_t::base;
if(k <= std::max<int64_t>(n, size(q.a))) {
return q.inv_inplace(k + n).div_xk_inplace(k);
}
if(k % 2) {
return inv_inplace(q, k - 1, n + 1).div_xk_inplace(1);
}
auto [q0, q1] = q.bisect();
auto qq = q0 * q0 - (q1 * q1).mul_xk_inplace(1);
inv_inplace(qq, k / 2 - q.deg() / 2, (n + 1) / 2 + q.deg() / 2);
size_t N = fft::com_size(size(q0.a), size(qq.a));
auto q0f = fft::dft<base>(q0.a, N);
auto q1f = fft::dft<base>(q1.a, N);
auto qqf = fft::dft<base>(qq.a, N);
size_t M = q0.deg() + (n + 1) / 2;
std::vector<base> A(M), B(M);
q0f.mul(qqf, A, M);
q1f.mul_inplace(qqf, B, M);
q.a.resize(n + 1);
for(size_t i = 0; i < n; i += 2) {
q.a[i] = A[q0.deg() + i / 2];
q.a[i + 1] = -B[q0.deg() + i / 2];
}
q.a.pop_back();
q.normalize();
return q;
}
template<typename poly>
poly& inv_inplace(poly& p, size_t n) {
using poly_t = std::decay_t<poly>;
using base = poly_t::base;
if(n == 1) {
return p = base(1) / p[0];
}
// Q(-x) = P0(x^2) + xP1(x^2)
auto [q0, q1] = p.bisect(n);
size_t N = fft::com_size(size(q0.a), (n + 1) / 2);
auto q0f = fft::dft<base>(q0.a, N);
auto q1f = fft::dft<base>(q1.a, N);
// Q(x)*Q(-x) = Q0(x^2)^2 - x^2 Q1(x^2)^2
auto qq = poly_t(q0f * q0f) - poly_t(q1f * q1f).mul_xk_inplace(1);
inv_inplace(qq, (n + 1) / 2);
auto qqf = fft::dft<base>(qq.a, N);
std::vector<base> A((n + 1) / 2), B((n + 1) / 2);
q0f.mul(qqf, A, (n + 1) / 2);
q1f.mul_inplace(qqf, B, (n + 1) / 2);
p.a.resize(n + 1);
for(size_t i = 0; i < n; i += 2) {
p.a[i] = A[i / 2];
p.a[i + 1] = -B[i / 2];
}
p.a.pop_back();
p.normalize();
return p;
}
}
#line 1 "cp-algo/math/combinatorics.hpp"
#line 5 "cp-algo/math/combinatorics.hpp"
namespace cp_algo::math {
// fact/rfact/small_inv are caching
// Beware of usage with dynamic mod
template<typename T>
T fact(int n) {
static std::vector<T> F(maxn);
static bool init = false;
if(!init) {
F[0] = T(1);
for(int i = 1; i < maxn; i++) {
F[i] = F[i - 1] * T(i);
}
init = true;
}
return F[n];
}
// Only works for modint types
template<typename T>
T rfact(int n) {
static std::vector<T> F(maxn);
static bool init = false;
if(!init) {
int t = (int)std::min<int64_t>(T::mod(), maxn) - 1;
F[t] = T(1) / fact<T>(t);
for(int i = t - 1; i >= 0; i--) {
F[i] = F[i + 1] * T(i + 1);
}
init = true;
}
return F[n];
}
template<typename T>
T small_inv(int n) {
static std::vector<T> F(maxn);
static bool init = false;
if(!init) {
for(int i = 1; i < maxn; i++) {
F[i] = rfact<T>(i) * fact<T>(i - 1);
}
init = true;
}
return F[n];
}
template<typename T>
T binom_large(T n, int r) {
assert(r < maxn);
T ans = 1;
for(int i = 0; i < r; i++) {
ans = ans * T(n - i) * small_inv<T>(i + 1);
}
return ans;
}
template<typename T>
T binom(int n, int r) {
if(r < 0 || r > n) {
return T(0);
} else if(n >= maxn) {
return binom_large(T(n), r);
} else {
return fact<T>(n) * rfact<T>(r) * rfact<T>(n - r);
}
}
}
#line 1 "cp-algo/number_theory/discrete_sqrt.hpp"
#line 1 "cp-algo/random/rng.hpp"
#include <chrono>
#include <random>
namespace cp_algo::random {
uint64_t rng() {
static std::mt19937_64 rng(
std::chrono::steady_clock::now().time_since_epoch().count()
);
return rng();
}
}
#line 6 "cp-algo/number_theory/discrete_sqrt.hpp"
namespace cp_algo::math {
// https://en.wikipedia.org/wiki/Berlekamp-Rabin_algorithm
template<modint_type base>
std::optional<base> sqrt(base b) {
if(b == base(0)) {
return base(0);
} else if(bpow(b, (b.mod() - 1) / 2) != base(1)) {
return std::nullopt;
} else {
while(true) {
base z = random::rng();
if(z * z == b) {
return z;
}
lin<base> x(1, z, b); // x + z (mod x^2 - b)
x = bpow(x, (b.mod() - 1) / 2, lin<base>(0, 1, b));
if(x.a != base(0)) {
return x.a.inv();
}
}
}
}
}
#line 16 "cp-algo/math/poly.hpp"
namespace cp_algo::math {
template<typename T>
struct poly_t {
using base = T;
std::vector<T> a;
void normalize() {poly::impl::normalize(*this);}
poly_t(){}
poly_t(T a0): a{a0} {normalize();}
poly_t(std::vector<T> const& t): a(t) {normalize();}
poly_t(std::vector<T>&& t): a(std::move(t)) {normalize();}
poly_t operator -() const {return poly::impl::neg_inplace(poly_t(*this));}
poly_t& operator += (poly_t const& t) {return poly::impl::add(*this, t);}
poly_t& operator -= (poly_t const& t) {return poly::impl::sub(*this, t);}
poly_t operator + (poly_t const& t) const {return poly_t(*this) += t;}
poly_t operator - (poly_t const& t) const {return poly_t(*this) -= t;}
poly_t& mod_xk_inplace(size_t k) {
a.resize(std::min(size(a), k));
normalize();
return *this;
}
poly_t& mul_xk_inplace(size_t k) {
a.insert(begin(a), k, T(0));
normalize();
return *this;
}
poly_t& div_xk_inplace(int64_t k) {
if(k < 0) {
return mul_xk_inplace(-k);
}
a.erase(begin(a), begin(a) + std::min<size_t>(k, size(a)));
normalize();
return *this;
}
poly_t mod_xk(size_t k) const {return poly_t(*this).mod_xk_inplace(k);}
poly_t mul_xk(size_t k) const {return poly_t(*this).mul_xk_inplace(k);}
poly_t div_xk(int64_t k) const {return poly_t(*this).div_xk_inplace(k);}
poly_t substr(size_t l, size_t k) const {return poly::impl::substr(*this, l, k);}
poly_t& operator *= (const poly_t &t) {fft::mul(a, t.a); normalize(); return *this;}
poly_t operator * (const poly_t &t) const {return poly_t(*this) *= t;}
poly_t& operator /= (const poly_t &t) {return *this = divmod(t)[0];}
poly_t& operator %= (const poly_t &t) {return *this = divmod(t)[1];}
poly_t operator / (poly_t const& t) const {return poly_t(*this) /= t;}
poly_t operator % (poly_t const& t) const {return poly_t(*this) %= t;}
poly_t& operator *= (T const& x) {return *this = poly::impl::scale(*this, x);}
poly_t& operator /= (T const& x) {return *this *= x.inv();}
poly_t operator * (T const& x) const {return poly_t(*this) *= x;}
poly_t operator / (T const& x) const {return poly_t(*this) /= x;}
poly_t& reverse(size_t n) {return poly::impl::reverse(*this, n);}
poly_t& reverse() {return reverse(size(a));}
poly_t reversed(size_t n) const {return poly_t(*this).reverse(n);}
poly_t reversed() const {return poly_t(*this).reverse();}
std::array<poly_t, 2> divmod(poly_t const& b) const {
return poly::impl::divmod(*this, b);
}
// reduces A/B to A'/B' such that
// deg B' < deg A / 2
static std::pair<std::list<poly_t>, linfrac<poly_t>> half_gcd(auto &&A, auto &&B) {
return poly::impl::half_gcd(A, B);
}
// reduces A / B to gcd(A, B) / 0
static std::pair<std::list<poly_t>, linfrac<poly_t>> full_gcd(auto &&A, auto &&B) {
return poly::impl::full_gcd(A, B);
}
static poly_t gcd(poly_t &&A, poly_t &&B) {
full_gcd(A, B);
return A;
}
// Returns a (non-monic) characteristic polynomial
// of the minimum linear recurrence for the sequence
poly_t min_rec(size_t d) const {
return poly::impl::min_rec(*this, d);
}
// calculate inv to *this modulo t
std::optional<poly_t> inv_mod(poly_t const& t) const {
return poly::impl::inv_mod(*this, t);
};
poly_t negx() const { // A(x) -> A(-x)
auto res = *this;
for(int i = 1; i <= deg(); i += 2) {
res.a[i] = -res[i];
}
return res;
}
void print(int n) const {
for(int i = 0; i < n; i++) {
std::cout << (*this)[i] << ' ';
}
std::cout << "\n";
}
void print() const {
print(deg() + 1);
}
T eval(T x) const { // evaluates in single point x
T res(0);
for(int i = deg(); i >= 0; i--) {
res *= x;
res += a[i];
}
return res;
}
T lead() const { // leading coefficient
assert(!is_zero());
return a.back();
}
int deg() const { // degree, -1 for P(x) = 0
return (int)a.size() - 1;
}
bool is_zero() const {
return a.empty();
}
T operator [](int idx) const {
return idx < 0 || idx > deg() ? T(0) : a[idx];
}
T& coef(size_t idx) { // mutable reference at coefficient
return a[idx];
}
bool operator == (const poly_t &t) const {return a == t.a;}
bool operator != (const poly_t &t) const {return a != t.a;}
poly_t& deriv_inplace(int k = 1) {
if(deg() + 1 < k) {
return *this = poly_t{};
}
for(int i = k; i <= deg(); i++) {
a[i - k] = fact<T>(i) * rfact<T>(i - k) * a[i];
}
a.resize(deg() + 1 - k);
return *this;
}
poly_t deriv(int k = 1) const { // calculate derivative
return poly_t(*this).deriv_inplace(k);
}
poly_t& integr_inplace() {
a.push_back(0);
for(int i = deg() - 1; i >= 0; i--) {
a[i + 1] = a[i] * small_inv<T>(i + 1);
}
a[0] = 0;
return *this;
}
poly_t integr() const { // calculate integral with C = 0
std::vector<T> res(deg() + 2);
for(int i = 0; i <= deg(); i++) {
res[i + 1] = a[i] * small_inv<T>(i + 1);
}
return res;
}
size_t trailing_xk() const { // Let p(x) = x^k * t(x), return k
if(is_zero()) {
return -1;
}
int res = 0;
while(a[res] == T(0)) {
res++;
}
return res;
}
// calculate log p(x) mod x^n
poly_t& log_inplace(size_t n) {
assert(a[0] == T(1));
mod_xk_inplace(n);
return (inv_inplace(n) *= mod_xk_inplace(n).deriv()).mod_xk_inplace(n - 1).integr_inplace();
}
poly_t log(size_t n) const {
return poly_t(*this).log_inplace(n);
}
poly_t& mul_truncate(poly_t const& t, size_t k) {
fft::mul_truncate(a, t.a, k);
normalize();
return *this;
}
poly_t& exp_inplace(size_t n) {
if(is_zero()) {
return *this = T(1);
}
assert(a[0] == T(0));
a[0] = 1;
size_t a = 1;
while(a < n) {
poly_t C = log(2 * a).div_xk_inplace(a) - substr(a, 2 * a);
*this -= C.mul_truncate(*this, a).mul_xk_inplace(a);
a *= 2;
}
return mod_xk_inplace(n);
}
poly_t exp(size_t n) const { // calculate exp p(x) mod x^n
return poly_t(*this).exp_inplace(n);
}
poly_t pow_bin(int64_t k, size_t n) const { // O(n log n log k)
if(k == 0) {
return poly_t(1).mod_xk(n);
} else {
auto t = pow(k / 2, n);
t = (t * t).mod_xk(n);
return (k % 2 ? *this * t : t).mod_xk(n);
}
}
poly_t circular_closure(size_t m) const {
if(deg() == -1) {
return *this;
}
auto t = *this;
for(size_t i = t.deg(); i >= m; i--) {
t.a[i - m] += t.a[i];
}
t.a.resize(std::min(t.a.size(), m));
return t;
}
static poly_t mul_circular(poly_t const& a, poly_t const& b, size_t m) {
return (a.circular_closure(m) * b.circular_closure(m)).circular_closure(m);
}
poly_t powmod_circular(int64_t k, size_t m) const {
if(k == 0) {
return poly_t(1);
} else {
auto t = powmod_circular(k / 2, m);
t = mul_circular(t, t, m);
if(k % 2) {
t = mul_circular(t, *this, m);
}
return t;
}
}
poly_t powmod(int64_t k, poly_t const& md) const {
return poly::impl::powmod(*this, k, md);
}
// O(d * n) with the derivative trick from
// https://codeforces.com/blog/entry/73947?#comment-581173
poly_t pow_dn(int64_t k, size_t n) const {
if(n == 0) {
return poly_t(T(0));
}
assert((*this)[0] != T(0));
std::vector<T> Q(n);
Q[0] = bpow(a[0], k);
auto a0inv = a[0].inv();
for(int i = 1; i < (int)n; i++) {
for(int j = 1; j <= std::min(deg(), i); j++) {
Q[i] += a[j] * Q[i - j] * (T(k) * T(j) - T(i - j));
}
Q[i] *= small_inv<T>(i) * a0inv;
}
return Q;
}
// calculate p^k(n) mod x^n in O(n log n)
// might be quite slow due to high constant
poly_t pow(int64_t k, size_t n) const {
if(is_zero()) {
return k ? *this : poly_t(1);
}
size_t i = trailing_xk();
if(i > 0) {
return k >= int64_t(n + i - 1) / (int64_t)i ? poly_t(T(0)) : div_xk(i).pow(k, n - i * k).mul_xk(i * k);
}
if(std::min(deg(), (int)n) <= magic) {
return pow_dn(k, n);
}
if(k <= magic) {
return pow_bin(k, n);
}
T j = a[i];
poly_t t = *this / j;
return bpow(j, k) * (t.log(n) * T(k)).exp(n).mod_xk(n);
}
// returns std::nullopt if undefined
std::optional<poly_t> sqrt(size_t n) const {
if(is_zero()) {
return *this;
}
size_t i = trailing_xk();
if(i % 2) {
return std::nullopt;
} else if(i > 0) {
auto ans = div_xk(i).sqrt(n - i / 2);
return ans ? ans->mul_xk(i / 2) : ans;
}
auto st = math::sqrt((*this)[0]);
if(st) {
poly_t ans = *st;
size_t a = 1;
while(a < n) {
a *= 2;
ans -= (ans - mod_xk(a) * ans.inv(a)).mod_xk(a) / 2;
}
return ans.mod_xk(n);
}
return std::nullopt;
}
poly_t mulx(T a) const { // component-wise multiplication with a^k
T cur = 1;
poly_t res(*this);
for(int i = 0; i <= deg(); i++) {
res.coef(i) *= cur;
cur *= a;
}
return res;
}
poly_t mulx_sq(T a) const { // component-wise multiplication with a^{k choose 2}
T cur = 1, total = 1;
poly_t res(*this);
for(int i = 0; i <= deg(); i++) {
res.coef(i) *= total;
cur *= a;
total *= cur;
}
return res;
}
// be mindful of maxn, as the function
// requires multiplying polynomials of size deg() and n+deg()!
poly_t chirpz(T z, int n) const { // P(1), P(z), P(z^2), ..., P(z^(n-1))
if(is_zero()) {
return std::vector<T>(n, 0);
}
if(z == T(0)) {
std::vector<T> ans(n, (*this)[0]);
if(n > 0) {
ans[0] = accumulate(begin(a), end(a), T(0));
}
return ans;
}
auto A = mulx_sq(z.inv());
auto B = ones(n+deg()).mulx_sq(z);
return semicorr(B, A).mod_xk(n).mulx_sq(z.inv());
}
// res[i] = prod_{1 <= j <= i} 1/(1 - z^j)
static auto _1mzk_prod_inv(T z, int n) {
std::vector<T> res(n, 1), zk(n);
zk[0] = 1;
for(int i = 1; i < n; i++) {
zk[i] = zk[i - 1] * z;
res[i] = res[i - 1] * (T(1) - zk[i]);
}
res.back() = res.back().inv();
for(int i = n - 2; i >= 0; i--) {
res[i] = (T(1) - zk[i+1]) * res[i+1];
}
return res;
}
// prod_{0 <= j < n} (1 - z^j x)
static auto _1mzkx_prod(T z, int n) {
if(n == 1) {
return poly_t(std::vector<T>{1, -1});
} else {
auto t = _1mzkx_prod(z, n / 2);
t *= t.mulx(bpow(z, n / 2));
if(n % 2) {
t *= poly_t(std::vector<T>{1, -bpow(z, n - 1)});
}
return t;
}
}
poly_t chirpz_inverse(T z, int n) const { // P(1), P(z), P(z^2), ..., P(z^(n-1))
if(is_zero()) {
return {};
}
if(z == T(0)) {
if(n == 1) {
return *this;
} else {
return std::vector{(*this)[1], (*this)[0] - (*this)[1]};
}
}
std::vector<T> y(n);
for(int i = 0; i < n; i++) {
y[i] = (*this)[i];
}
auto prods_pos = _1mzk_prod_inv(z, n);
auto prods_neg = _1mzk_prod_inv(z.inv(), n);
T zn = bpow(z, n-1).inv();
T znk = 1;
for(int i = 0; i < n; i++) {
y[i] *= znk * prods_neg[i] * prods_pos[(n - 1) - i];
znk *= zn;
}
poly_t p_over_q = poly_t(y).chirpz(z, n);
poly_t q = _1mzkx_prod(z, n);
return (p_over_q * q).mod_xk_inplace(n).reverse(n);
}
static poly_t build(std::vector<poly_t> &res, int v, auto L, auto R) { // builds evaluation tree for (x-a1)(x-a2)...(x-an)
if(R - L == 1) {
return res[v] = std::vector<T>{-*L, 1};
} else {
auto M = L + (R - L) / 2;
return res[v] = build(res, 2 * v, L, M) * build(res, 2 * v + 1, M, R);
}
}
poly_t to_newton(std::vector<poly_t> &tree, int v, auto l, auto r) {
if(r - l == 1) {
return *this;
} else {
auto m = l + (r - l) / 2;
auto A = (*this % tree[2 * v]).to_newton(tree, 2 * v, l, m);
auto B = (*this / tree[2 * v]).to_newton(tree, 2 * v + 1, m, r);
return A + B.mul_xk(m - l);
}
}
poly_t to_newton(std::vector<T> p) {
if(is_zero()) {
return *this;
}
size_t n = p.size();
std::vector<poly_t> tree(4 * n);
build(tree, 1, begin(p), end(p));
return to_newton(tree, 1, begin(p), end(p));
}
std::vector<T> eval(std::vector<poly_t> &tree, int v, auto l, auto r) { // auxiliary evaluation function
if(r - l == 1) {
return {eval(*l)};
} else {
auto m = l + (r - l) / 2;
auto A = (*this % tree[2 * v]).eval(tree, 2 * v, l, m);
auto B = (*this % tree[2 * v + 1]).eval(tree, 2 * v + 1, m, r);
A.insert(end(A), begin(B), end(B));
return A;
}
}
std::vector<T> eval(std::vector<T> x) { // evaluate polynomial in (x1, ..., xn)
size_t n = x.size();
if(is_zero()) {
return std::vector<T>(n, T(0));
}
std::vector<poly_t> tree(4 * n);
build(tree, 1, begin(x), end(x));
return eval(tree, 1, begin(x), end(x));
}
poly_t inter(std::vector<poly_t> &tree, int v, auto ly, auto ry) { // auxiliary interpolation function
if(ry - ly == 1) {
return {*ly / a[0]};
} else {
auto my = ly + (ry - ly) / 2;
auto A = (*this % tree[2 * v]).inter(tree, 2 * v, ly, my);
auto B = (*this % tree[2 * v + 1]).inter(tree, 2 * v + 1, my, ry);
return A * tree[2 * v + 1] + B * tree[2 * v];
}
}
static auto inter(std::vector<T> x, std::vector<T> y) { // interpolates minimum polynomial from (xi, yi) pairs
size_t n = x.size();
std::vector<poly_t> tree(4 * n);
return build(tree, 1, begin(x), end(x)).deriv().inter(tree, 1, begin(y), end(y));
}
static auto resultant(poly_t a, poly_t b) { // computes resultant of a and b
if(b.is_zero()) {
return 0;
} else if(b.deg() == 0) {
return bpow(b.lead(), a.deg());
} else {
int pw = a.deg();
a %= b;
pw -= a.deg();
auto mul = bpow(b.lead(), pw) * T((b.deg() & a.deg() & 1) ? -1 : 1);
auto ans = resultant(b, a);
return ans * mul;
}
}
static poly_t xk(size_t n) { // P(x) = x^n
return poly_t(T(1)).mul_xk(n);
}
static poly_t ones(size_t n) { // P(x) = 1 + x + ... + x^{n-1}
return std::vector<T>(n, 1);
}
static poly_t expx(size_t n) { // P(x) = e^x (mod x^n)
return ones(n).borel();
}
static poly_t log1px(size_t n) { // P(x) = log(1+x) (mod x^n)
std::vector<T> coeffs(n, 0);
for(size_t i = 1; i < n; i++) {
coeffs[i] = (i & 1 ? T(i).inv() : -T(i).inv());
}
return coeffs;
}
static poly_t log1mx(size_t n) { // P(x) = log(1-x) (mod x^n)
return -ones(n).integr();
}
// [x^k] (a corr b) = sum_{i} a{(k-m)+i}*bi
static poly_t corr(poly_t const& a, poly_t const& b) { // cross-correlation
return a * b.reversed();
}
// [x^k] (a semicorr b) = sum_i a{i+k} * b{i}
static poly_t semicorr(poly_t const& a, poly_t const& b) {
return corr(a, b).div_xk(b.deg());
}
poly_t invborel() const { // ak *= k!
auto res = *this;
for(int i = 0; i <= deg(); i++) {
res.coef(i) *= fact<T>(i);
}
return res;
}
poly_t borel() const { // ak /= k!
auto res = *this;
for(int i = 0; i <= deg(); i++) {
res.coef(i) *= rfact<T>(i);
}
return res;
}
poly_t shift(T a) const { // P(x + a)
return semicorr(invborel(), expx(deg() + 1).mulx(a)).borel();
}
poly_t x2() { // P(x) -> P(x^2)
std::vector<T> res(2 * a.size());
for(size_t i = 0; i < a.size(); i++) {
res[2 * i] = a[i];
}
return res;
}
// Return {P0, P1}, where P(x) = P0(x) + xP1(x)
std::array<poly_t, 2> bisect(size_t n) const {
n = std::min(n, size(a));
std::vector<T> res[2];
for(size_t i = 0; i < n; i++) {
res[i % 2].push_back(a[i]);
}
return {res[0], res[1]};
}
std::array<poly_t, 2> bisect() const {
return bisect(size(a));
}
// Find [x^k] P / Q
static T kth_rec_inplace(poly_t &P, poly_t &Q, int64_t k) {
while(k > Q.deg()) {
size_t n = Q.a.size();
auto [Q0, Q1] = Q.bisect();
auto [P0, P1] = P.bisect();
size_t N = fft::com_size((n + 1) / 2, (n + 1) / 2);
auto Q0f = fft::dft<T>(Q0.a, N);
auto Q1f = fft::dft<T>(Q1.a, N);
auto P0f = fft::dft<T>(P0.a, N);
auto P1f = fft::dft<T>(P1.a, N);
Q = poly_t(Q0f * Q0f) -= poly_t(Q1f * Q1f).mul_xk_inplace(1);
if(k % 2) {
P = poly_t(Q0f *= P1f) -= poly_t(Q1f *= P0f);
} else {
P = poly_t(Q0f *= P0f) -= poly_t(Q1f *= P1f).mul_xk_inplace(1);
}
k /= 2;
}
return (P *= Q.inv_inplace(Q.deg() + 1))[(int)k];
}
static T kth_rec(poly_t const& P, poly_t const& Q, int64_t k) {
return kth_rec_inplace(poly_t(P), poly_t(Q), k);
}
// inverse series mod x^n
poly_t& inv_inplace(size_t n) {
return poly::impl::inv_inplace(*this, n);
}
poly_t inv(size_t n) const {
return poly_t(*this).inv_inplace(n);
}
// [x^k]..[x^{k+n-1}] of inv()
// supports negative k if k+n >= 0
poly_t& inv_inplace(int64_t k, size_t n) {
return poly::impl::inv_inplace(*this, k, n);
}
poly_t inv(int64_t k, size_t n) const {
return poly_t(*this).inv_inplace(k, n);
}
// compute A(B(x)) mod x^n in O(n^2)
static poly_t compose(poly_t A, poly_t B, int n) {
int q = std::sqrt(n);
std::vector<poly_t> Bk(q);
auto Bq = B.pow(q, n);
Bk[0] = poly_t(T(1));
for(int i = 1; i < q; i++) {
Bk[i] = (Bk[i - 1] * B).mod_xk(n);
}
poly_t Bqk(1);
poly_t ans;
for(int i = 0; i <= n / q; i++) {
poly_t cur;
for(int j = 0; j < q; j++) {
cur += Bk[j] * A[i * q + j];
}
ans += (Bqk * cur).mod_xk(n);
Bqk = (Bqk * Bq).mod_xk(n);
}
return ans;
}
// compute A(B(x)) mod x^n in O(sqrt(pqn log^3 n))
// preferrable when p = deg A and q = deg B
// are much less than n
static poly_t compose_large(poly_t A, poly_t B, int n) {
if(B[0] != T(0)) {
return compose_large(A.shift(B[0]), B - B[0], n);
}
int q = std::sqrt(n);
auto [B0, B1] = std::make_pair(B.mod_xk(q), B.div_xk(q));
B0 = B0.div_xk(1);
std::vector<poly_t> pw(A.deg() + 1);
auto getpow = [&](int k) {
return pw[k].is_zero() ? pw[k] = B0.pow(k, n - k) : pw[k];
};
std::function<poly_t(poly_t const&, int, int)> compose_dac = [&getpow, &compose_dac](poly_t const& f, int m, int N) {
if(f.deg() <= 0) {
return f;
}
int k = m / 2;
auto [f0, f1] = std::make_pair(f.mod_xk(k), f.div_xk(k));
auto [A, B] = std::make_pair(compose_dac(f0, k, N), compose_dac(f1, m - k, N - k));
return (A + (B.mod_xk(N - k) * getpow(k).mod_xk(N - k)).mul_xk(k)).mod_xk(N);
};
int r = n / q;
auto Ar = A.deriv(r);
auto AB0 = compose_dac(Ar, Ar.deg() + 1, n);
auto Bd = B0.mul_xk(1).deriv();
poly_t ans = T(0);
std::vector<poly_t> B1p(r + 1);
B1p[0] = poly_t(T(1));
for(int i = 1; i <= r; i++) {
B1p[i] = (B1p[i - 1] * B1.mod_xk(n - i * q)).mod_xk(n - i * q);
}
while(r >= 0) {
ans += (AB0.mod_xk(n - r * q) * rfact<T>(r) * B1p[r]).mul_xk(r * q).mod_xk(n);
r--;
if(r >= 0) {
AB0 = ((AB0 * Bd).integr() + A[r] * fact<T>(r)).mod_xk(n);
}
}
return ans;
}
};
template<typename base>
static auto operator * (const auto& a, const poly_t<base>& b) {
return b * a;
}
};
#line 5 "verify/poly/bell.test.cpp"
#include <bits/stdc++.h>
using namespace std;
using namespace cp_algo::math;
const int mod = 998244353;
using base = modint<mod>;
using polyn = poly_t<base>;
void solve() {
int n;
cin >> n;
(polyn::expx(n+1) - polyn(1)).exp_inplace(n+1).invborel().print(n+1);
}
signed main() {
//freopen("input.txt", "r", stdin);
ios::sync_with_stdio(0);
cin.tie(0);
int t = 1;
while(t--) {
solve();
}
}
Test cases
| Env | Name | Status | Elapsed | Memory |
|---|---|---|---|---|
| g++ | 0_00 |
AC |
10 ms | 10 MB |
| g++ | 100000_00 |
AC |
153 ms | 26 MB |
| g++ | 10000_00 |
AC |
28 ms | 13 MB |
| g++ | 1000_00 |
AC |
14 ms | 12 MB |
| g++ | 100_00 |
AC |
12 ms | 12 MB |
| g++ | 1_00 |
AC |
12 ms | 12 MB |
| g++ | 200000_00 |
AC |
269 ms | 40 MB |
| g++ | 300000_00 |
AC |
528 ms | 65 MB |
| g++ | 400000_00 |
AC |
560 ms | 63 MB |
| g++ | 500000_00 |
AC |
549 ms | 67 MB |
| g++ | example_00 |
AC |
13 ms | 12 MB |