結果
| 問題 | No.1145 Sums of Powers |
| ユーザー |
 suisen
|
| 提出日時 | 2023-01-25 18:20:38 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 404 ms / 2,000 ms |
| コード長 | 43,032 bytes |
| コンパイル時間 | 3,737 ms |
| コンパイル使用メモリ | 160,852 KB |
| 実行使用メモリ | 25,740 KB |
| 最終ジャッジ日時 | 2024-06-26 21:28:30 |
| 合計ジャッジ時間 | 5,521 ms |
|
ジャッジサーバーID （参考情報） |
judge3 / judge4 |
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テストケース
テストケース表示| 入力 | 結果 | 実行時間 実行使用メモリ |
|---|---|---|
| testcase_00 | AC | 2 ms
5,248 KB |
| testcase_01 | AC | 2 ms
5,376 KB |
| testcase_02 | AC | 4 ms
5,376 KB |
| testcase_03 | AC | 404 ms
25,740 KB |
| testcase_04 | AC | 394 ms
25,484 KB |
| testcase_05 | AC | 402 ms
25,588 KB |
ソースコード
#include <iostream>
#include <atcoder/modint>
using mint = atcoder::modint998244353;
std::istream& operator>>(std::istream& in, mint &a) {
long long e; in >> e; a = e;
return in;
}
std::ostream& operator<<(std::ostream& out, const mint &a) {
out << a.val();
return out;
}
#include <limits>
#include <optional>
#include <queue>
#include <atcoder/modint>
#include <atcoder/convolution>
#include <cassert>
#include <cmath>
#include <type_traits>
#include <vector>
namespace suisen {
// ! utility
template <typename ...Types>
using constraints_t = std::enable_if_t<std::conjunction_v<Types...>, std::nullptr_t>;
template <bool cond_v, typename Then, typename OrElse>
constexpr decltype(auto) constexpr_if(Then&& then, OrElse&& or_else) {
if constexpr (cond_v) {
return std::forward<Then>(then);
} else {
return std::forward<OrElse>(or_else);
}
}
// ! function
template <typename ReturnType, typename Callable, typename ...Args>
using is_same_as_invoke_result = std::is_same<std::invoke_result_t<Callable, Args...>, ReturnType>;
template <typename F, typename T>
using is_uni_op = is_same_as_invoke_result<T, F, T>;
template <typename F, typename T>
using is_bin_op = is_same_as_invoke_result<T, F, T, T>;
template <typename Comparator, typename T>
using is_comparator = std::is_same<std::invoke_result_t<Comparator, T, T>, bool>;
// ! integral
template <typename T, typename = constraints_t<std::is_integral<T>>>
constexpr int bit_num = std::numeric_limits<std::make_unsigned_t<T>>::digits;
template <typename T, unsigned int n>
struct is_nbit { static constexpr bool value = bit_num<T> == n; };
template <typename T, unsigned int n>
static constexpr bool is_nbit_v = is_nbit<T, n>::value;
// ?
template <typename T>
struct safely_multipliable {};
template <>
struct safely_multipliable<int> { using type = long long; };
template <>
struct safely_multipliable<long long> { using type = __int128_t; };
template <>
struct safely_multipliable<unsigned int> { using type = unsigned long long; };
template <>
struct safely_multipliable<unsigned long int> { using type = __uint128_t; };
template <>
struct safely_multipliable<unsigned long long> { using type = __uint128_t; };
template <>
struct safely_multipliable<float> { using type = float; };
template <>
struct safely_multipliable<double> { using type = double; };
template <>
struct safely_multipliable<long double> { using type = long double; };
template <typename T>
using safely_multipliable_t = typename safely_multipliable<T>::type;
template <typename T, typename = void>
struct rec_value_type {
using type = T;
};
template <typename T>
struct rec_value_type<T, std::void_t<typename T::value_type>> {
using type = typename rec_value_type<typename T::value_type>::type;
};
template <typename T>
using rec_value_type_t = typename rec_value_type<T>::type;
} // namespace suisen
/**
* refernce: https://37zigen.com/tonelli-shanks-algorithm/
* calculates x s.t. x^2 = a mod p in O((log p)^2).
*/
template <typename mint>
std::optional<mint> safe_sqrt(mint a) {
static int p = mint::mod();
if (a == 0) return std::make_optional(0);
if (p == 2) return std::make_optional(a);
if (a.pow((p - 1) / 2) != 1) return std::nullopt;
mint b = 1;
while (b.pow((p - 1) / 2) == 1) ++b;
static int tlz = __builtin_ctz(p - 1), q = (p - 1) >> tlz;
mint x = a.pow((q + 1) / 2);
b = b.pow(q);
for (int shift = 2; x * x != a; ++shift) {
mint e = a.inv() * x * x;
if (e.pow(1 << (tlz - shift)) != 1) x *= b;
b *= b;
}
return std::make_optional(x);
}
/**
* calculates x s.t. x^2 = a mod p in O((log p)^2).
* if not exists, raises runtime error.
*/
template <typename mint>
auto sqrt(mint a) -> decltype(mint::mod(), mint()) {
return *safe_sqrt(a);
}
template <typename mint>
auto log(mint a) -> decltype(mint::mod(), mint()) {
assert(a == 1);
return 0;
}
template <typename mint>
auto exp(mint a) -> decltype(mint::mod(), mint()) {
assert(a == 0);
return 1;
}
template <typename mint, typename T>
auto pow(mint a, T b) -> decltype(mint::mod(), mint()) {
return a.pow(b);
}
template <typename mint>
auto inv(mint a) -> decltype(mint::mod(), mint()) {
return a.inv();
}
namespace suisen {
template <typename mint>
class inv_mods {
public:
inv_mods() {}
inv_mods(int n) { ensure(n); }
const mint& operator[](int i) const {
ensure(i);
return invs[i];
}
static void ensure(int n) {
int sz = invs.size();
if (sz < 2) invs = { 0, 1 }, sz = 2;
if (sz < n + 1) {
invs.resize(n + 1);
for (int i = sz; i <= n; ++i) invs[i] = mint(mod - mod / i) * invs[mod % i];
}
}
private:
static std::vector<mint> invs;
static constexpr int mod = mint::mod();
};
template <typename mint>
std::vector<mint> inv_mods<mint>::invs{};
template <typename mint>
std::vector<mint> get_invs(const std::vector<mint>& vs) {
const int n = vs.size();
mint p = 1;
for (auto& e : vs) {
p *= e;
assert(e != 0);
}
mint ip = p.inv();
std::vector<mint> rp(n + 1);
rp[n] = 1;
for (int i = n - 1; i >= 0; --i) {
rp[i] = rp[i + 1] * vs[i];
}
std::vector<mint> res(n);
for (int i = 0; i < n; ++i) {
res[i] = ip * rp[i + 1];
ip *= vs[i];
}
return res;
}
}
namespace suisen {
template <typename T>
struct FPSNaive : std::vector<T> {
static inline int MAX_SIZE = std::numeric_limits<int>::max() / 2;
using value_type = T;
using element_type = rec_value_type_t<T>;
using std::vector<value_type>::vector;
FPSNaive(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
FPSNaive(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}
static void set_max_size(int n) {
FPSNaive<T>::MAX_SIZE = n;
}
const value_type operator[](int n) const {
return n <= deg() ? unsafe_get(n) : value_type{ 0 };
}
value_type& operator[](int n) {
return ensure_deg(n), unsafe_get(n);
}
int size() const {
return std::vector<value_type>::size();
}
int deg() const {
return size() - 1;
}
int normalize() {
while (size() and this->back() == value_type{ 0 }) this->pop_back();
return deg();
}
FPSNaive& cut_inplace(int n) {
if (size() > n) this->resize(std::max(0, n));
return *this;
}
FPSNaive cut(int n) const {
FPSNaive f = FPSNaive(*this).cut_inplace(n);
return f;
}
FPSNaive operator+() const {
return FPSNaive(*this);
}
FPSNaive operator-() const {
FPSNaive f(*this);
for (auto& e : f) e = -e;
return f;
}
FPSNaive& operator++() { return ++(*this)[0], * this; }
FPSNaive& operator--() { return --(*this)[0], * this; }
FPSNaive& operator+=(const value_type x) { return (*this)[0] += x, *this; }
FPSNaive& operator-=(const value_type x) { return (*this)[0] -= x, *this; }
FPSNaive& operator+=(const FPSNaive& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) += g.unsafe_get(i);
return *this;
}
FPSNaive& operator-=(const FPSNaive& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) -= g.unsafe_get(i);
return *this;
}
FPSNaive& operator*=(const FPSNaive& g) { return *this = *this * g; }
FPSNaive& operator*=(const value_type x) {
for (auto& e : *this) e *= x;
return *this;
}
FPSNaive& operator/=(const FPSNaive& g) { return *this = *this / g; }
FPSNaive& operator%=(const FPSNaive& g) { return *this = *this % g; }
FPSNaive& operator<<=(const int shamt) {
this->insert(this->begin(), shamt, value_type{ 0 });
return *this;
}
FPSNaive& operator>>=(const int shamt) {
if (shamt > size()) this->clear();
else this->erase(this->begin(), this->begin() + shamt);
return *this;
}
friend FPSNaive operator+(FPSNaive f, const FPSNaive& g) { f += g; return f; }
friend FPSNaive operator+(FPSNaive f, const value_type& x) { f += x; return f; }
friend FPSNaive operator-(FPSNaive f, const FPSNaive& g) { f -= g; return f; }
friend FPSNaive operator-(FPSNaive f, const value_type& x) { f -= x; return f; }
friend FPSNaive operator*(const FPSNaive& f, const FPSNaive& g) {
if (f.empty() or g.empty()) return FPSNaive{};
const int n = f.size(), m = g.size();
FPSNaive h(std::min(MAX_SIZE, n + m - 1));
for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) {
if (i + j >= MAX_SIZE) break;
h.unsafe_get(i + j) += f.unsafe_get(i) * g.unsafe_get(j);
}
return h;
}
friend FPSNaive operator*(FPSNaive f, const value_type& x) { f *= x; return f; }
friend FPSNaive operator/(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).first); }
friend FPSNaive operator%(FPSNaive f, const FPSNaive& g) { return std::move(f.div_mod(g).second); }
friend FPSNaive operator*(const value_type x, FPSNaive f) { f *= x; return f; }
friend FPSNaive operator<<(FPSNaive f, const int shamt) { f <<= shamt; return f; }
friend FPSNaive operator>>(FPSNaive f, const int shamt) { f >>= shamt; return f; }
std::pair<FPSNaive, FPSNaive> div_mod(FPSNaive g) const {
FPSNaive f = *this;
const int fd = f.normalize(), gd = g.normalize();
assert(gd >= 0);
if (fd < gd) return { FPSNaive{}, f };
if (gd == 0) return { f *= g.unsafe_get(0).inv(), FPSNaive{} };
const int k = f.deg() - gd;
value_type head_inv = g.unsafe_get(gd).inv();
FPSNaive q(k + 1);
for (int i = k; i >= 0; --i) {
value_type div = f.unsafe_get(i + gd) * head_inv;
q.unsafe_get(i) = div;
for (int j = 0; j <= gd; ++j) f.unsafe_get(i + j) -= div * g.unsafe_get(j);
}
return { q, f.cut_inplace(gd) };
}
friend bool operator==(const FPSNaive& f, const FPSNaive& g) {
const int n = f.size(), m = g.size();
if (n < m) return g == f;
for (int i = 0; i < m; ++i) if (f.unsafe_get(i) != g.unsafe_get(i)) return false;
for (int i = m; i < n; ++i) if (f.unsafe_get(i) != 0) return false;
return true;
}
friend bool operator!=(const FPSNaive& f, const FPSNaive& g) {
return not (f == g);
}
FPSNaive mul(const FPSNaive& g, int n = -1) const {
if (n < 0) n = size();
if (this->empty() or g.empty()) return FPSNaive{};
const int m = size(), k = g.size();
FPSNaive h(std::min(n, m + k - 1));
for (int i = 0; i < m; ++i) {
for (int j = 0, jr = std::min(k, n - i); j < jr; ++j) {
h.unsafe_get(i + j) += unsafe_get(i) * g.unsafe_get(j);
}
}
return h;
}
FPSNaive diff() const {
if (this->empty()) return {};
FPSNaive g(size() - 1);
for (int i = 1; i <= deg(); ++i) g.unsafe_get(i - 1) = unsafe_get(i) * i;
return g;
}
FPSNaive intg() const {
const int n = size();
FPSNaive g(n + 1);
for (int i = 0; i < n; ++i) g.unsafe_get(i + 1) = unsafe_get(i) * invs[i + 1];
if (g.deg() > MAX_SIZE) g.cut_inplace(MAX_SIZE);
return g;
}
FPSNaive inv(int n = -1) const {
if (n < 0) n = size();
FPSNaive g(n);
const value_type inv_f0 = ::inv(unsafe_get(0));
g.unsafe_get(0) = inv_f0;
for (int i = 1; i < n; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) -= g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= inv_f0;
}
return g;
}
FPSNaive exp(int n = -1) const {
if (n < 0) n = size();
assert(unsafe_get(0) == value_type{ 0 });
FPSNaive g(n);
g.unsafe_get(0) = value_type{ 1 };
for (int i = 1; i < n; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) += j * g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= invs[i];
}
return g;
}
FPSNaive log(int n = -1) const {
if (n < 0) n = size();
assert(unsafe_get(0) == value_type{ 1 });
FPSNaive g(n);
g.unsafe_get(0) = value_type{ 0 };
for (int i = 1; i < n; ++i) {
g.unsafe_get(i) = i * (*this)[i];
for (int j = 1; j < i; ++j) g.unsafe_get(i) -= (i - j) * g.unsafe_get(i - j) * (*this)[j];
g.unsafe_get(i) *= invs[i];
}
return g;
}
FPSNaive pow(const long long k, int n = -1) const {
if (n < 0) n = size();
if (k == 0) {
FPSNaive res(n);
res[0] = 1;
return res;
}
int z = 0;
while (z < size() and unsafe_get(z) == value_type{ 0 }) ++z;
if (z == size() or z > (n - 1) / k) return FPSNaive(n, 0);
const int m = n - z * k;
FPSNaive g(m);
const value_type inv_f0 = ::inv(unsafe_get(z));
g.unsafe_get(0) = unsafe_get(z).pow(k);
for (int i = 1; i < m; ++i) {
for (int j = 1; j <= i; ++j) g.unsafe_get(i) += (element_type{ k } *j - (i - j)) * g.unsafe_get(i - j) * (*this)[z + j];
g.unsafe_get(i) *= inv_f0 * invs[i];
}
g <<= z * k;
return g;
}
std::optional<FPSNaive> safe_sqrt(int n = -1) const {
if (n < 0) n = size();
int dl = 0;
while (dl < size() and unsafe_get(dl) == value_type{ 0 }) ++dl;
if (dl == size()) return FPSNaive(n, 0);
if (dl & 1) return std::nullopt;
const int m = n - dl / 2;
FPSNaive g(m);
auto opt_g0 = ::safe_sqrt((*this)[dl]);
if (not opt_g0.has_value()) return std::nullopt;
g.unsafe_get(0) = *opt_g0;
value_type inv_2g0 = ::inv(2 * g.unsafe_get(0));
for (int i = 1; i < m; ++i) {
g.unsafe_get(i) = (*this)[dl + i];
for (int j = 1; j < i; ++j) g.unsafe_get(i) -= g.unsafe_get(j) * g.unsafe_get(i - j);
g.unsafe_get(i) *= inv_2g0;
}
g <<= dl / 2;
return g;
}
FPSNaive sqrt(int n = -1) const {
if (n < 0) n = size();
return *safe_sqrt(n);
}
value_type eval(value_type x) const {
value_type y = 0;
for (int i = size() - 1; i >= 0; --i) y = y * x + unsafe_get(i);
return y;
}
private:
static inline inv_mods<element_type> invs;
void ensure_deg(int d) {
if (deg() < d) this->resize(d + 1, value_type{ 0 });
}
const value_type& unsafe_get(int i) const {
return std::vector<value_type>::operator[](i);
}
value_type& unsafe_get(int i) {
return std::vector<value_type>::operator[](i);
}
};
} // namespace suisen
template <typename mint>
suisen::FPSNaive<mint> sqrt(suisen::FPSNaive<mint> a) {
return a.sqrt();
}
template <typename mint>
suisen::FPSNaive<mint> log(suisen::FPSNaive<mint> a) {
return a.log();
}
template <typename mint>
suisen::FPSNaive<mint> exp(suisen::FPSNaive<mint> a) {
return a.exp();
}
template <typename mint, typename T>
suisen::FPSNaive<mint> pow(suisen::FPSNaive<mint> a, T b) {
return a.pow(b);
}
template <typename mint>
suisen::FPSNaive<mint> inv(suisen::FPSNaive<mint> a) {
return a.inv();
}
namespace suisen {
template <typename mint, atcoder::internal::is_static_modint_t<mint>* = nullptr>
struct FormalPowerSeries : std::vector<mint> {
using base_type = std::vector<mint>;
using value_type = typename base_type::value_type;
using base_type::vector;
FormalPowerSeries(const std::initializer_list<value_type> l) : std::vector<value_type>::vector(l) {}
FormalPowerSeries(const std::vector<value_type>& v) : std::vector<value_type>::vector(v) {}
int size() const noexcept {
return base_type::size();
}
int deg() const noexcept {
return size() - 1;
}
void ensure(int n) {
if (size() < n) this->resize(n);
}
value_type safe_get(int d) const {
return d <= deg() ? (*this)[d] : 0;
}
value_type& safe_get(int d) {
ensure(d + 1);
return (*this)[d];
}
FormalPowerSeries& cut_trailing_zeros() {
while (size() and this->back() == 0) this->pop_back();
return *this;
}
FormalPowerSeries& cut(int n) {
if (size() > n) this->resize(std::max(0, n));
return *this;
}
FormalPowerSeries cut_copy(int n) const {
FormalPowerSeries res(this->begin(), this->begin() + std::min(size(), n));
res.ensure(n);
return res;
}
FormalPowerSeries cut_copy(int l, int r) const {
if (l >= size()) return FormalPowerSeries(r - l, 0);
FormalPowerSeries res(this->begin() + l, this->begin() + std::min(size(), r));
res.ensure(r - l);
return res;
}
/* Unary Operations */
FormalPowerSeries operator+() const { return *this; }
FormalPowerSeries operator-() const {
FormalPowerSeries res = *this;
for (auto& e : res) e = -e;
return res;
}
FormalPowerSeries& operator++() { return ++safe_get(0), * this; }
FormalPowerSeries& operator--() { return --safe_get(0), * this; }
FormalPowerSeries operator++(int) {
FormalPowerSeries res = *this;
++(*this);
return res;
}
FormalPowerSeries operator--(int) {
FormalPowerSeries res = *this;
--(*this);
return res;
}
/* Binary Operations With Constant */
FormalPowerSeries& operator+=(const value_type& x) { return safe_get(0) += x, *this; }
FormalPowerSeries& operator-=(const value_type& x) { return safe_get(0) -= x, *this; }
FormalPowerSeries& operator*=(const value_type& x) {
for (auto& e : *this) e *= x;
return *this;
}
FormalPowerSeries& operator/=(const value_type& x) { return *this *= x.inv(); }
friend FormalPowerSeries operator+(FormalPowerSeries f, const value_type& x) { f += x; return f; }
friend FormalPowerSeries operator+(const value_type& x, FormalPowerSeries f) { f += x; return f; }
friend FormalPowerSeries operator-(FormalPowerSeries f, const value_type& x) { f -= x; return f; }
friend FormalPowerSeries operator-(const value_type& x, FormalPowerSeries f) { f -= x; return -f; }
friend FormalPowerSeries operator*(FormalPowerSeries f, const value_type& x) { f *= x; return f; }
friend FormalPowerSeries operator*(const value_type& x, FormalPowerSeries f) { f *= x; return f; }
friend FormalPowerSeries operator/(FormalPowerSeries f, const value_type& x) { f /= x; return f; }
/* Binary Operations With Formal Power Series */
FormalPowerSeries& operator+=(const FormalPowerSeries& g) {
const int n = g.size();
ensure(n);
for (int i = 0; i < n; ++i) (*this)[i] += g[i];
return *this;
}
FormalPowerSeries& operator-=(const FormalPowerSeries& g) {
const int n = g.size();
ensure(n);
for (int i = 0; i < n; ++i) (*this)[i] -= g[i];
return *this;
}
FormalPowerSeries& operator*=(const FormalPowerSeries& g) { return *this = *this * g; }
FormalPowerSeries& operator/=(const FormalPowerSeries& g) { return *this = *this / g; }
FormalPowerSeries& operator%=(const FormalPowerSeries& g) { return *this = *this % g; }
friend FormalPowerSeries operator+(FormalPowerSeries f, const FormalPowerSeries& g) { f += g; return f; }
friend FormalPowerSeries operator-(FormalPowerSeries f, const FormalPowerSeries& g) { f -= g; return f; }
friend FormalPowerSeries operator*(const FormalPowerSeries& f, const FormalPowerSeries& g) {
const int siz_f = f.size(), siz_g = g.size();
if (siz_f < siz_g) return g * f;
if (std::min(siz_f, siz_g) <= 60) return atcoder::convolution(f, g);
const int deg = siz_f + siz_g - 2;
int fpow2 = 1;
while ((fpow2 << 1) <= deg) fpow2 <<= 1;
if (const int dif = deg - fpow2 + 1; dif <= 10) {
FormalPowerSeries h = atcoder::convolution(std::vector<mint>(f.begin(), f.end() - dif), g);
h.resize(h.size() + dif);
for (int i = siz_f - dif; i < siz_f; ++i) for (int j = 0; j < siz_g; ++j) {
h[i + j] += f[i] * g[j];
}
return h;
}
return atcoder::convolution(f, g);
}
friend FormalPowerSeries operator/(FormalPowerSeries f, FormalPowerSeries g) {
if (f.size() < 60) return FPSNaive<mint>(f).div_mod(g).first;
f.cut_trailing_zeros(), g.cut_trailing_zeros();
const int fd = f.deg(), gd = g.deg();
assert(gd >= 0);
if (fd < gd) return {};
if (gd == 0) {
f /= g[0];
return f;
}
std::reverse(f.begin(), f.end()), std::reverse(g.begin(), g.end());
const int qd = fd - gd;
FormalPowerSeries q = f * g.inv(qd + 1);
q.cut(qd + 1);
std::reverse(q.begin(), q.end());
return q;
}
friend FormalPowerSeries operator%(const FormalPowerSeries& f, const FormalPowerSeries& g) { return f.div_mod(g).second; }
std::pair<FormalPowerSeries, FormalPowerSeries> div_mod(const FormalPowerSeries& g) const {
if (size() < 60) {
auto [q, r] = FPSNaive<mint>(*this).div_mod(g);
return { q, r };
}
FormalPowerSeries q = *this / g, r = *this - g * q;
r.cut_trailing_zeros();
return { q, r };
}
/* Shift Operations */
FormalPowerSeries& operator<<=(const int shamt) {
return this->insert(this->begin(), shamt, 0), * this;
}
FormalPowerSeries& operator>>=(const int shamt) {
return this->erase(this->begin(), this->begin() + std::min(shamt, size())), * this;
}
friend FormalPowerSeries operator<<(FormalPowerSeries f, const int shamt) { f <<= shamt; return f; }
friend FormalPowerSeries operator>>(FormalPowerSeries f, const int shamt) { f >>= shamt; return f; }
/* Compare */
friend bool operator==(const FormalPowerSeries& f, const FormalPowerSeries& g) {
const int n = f.size(), m = g.size();
if (n < m) return g == f;
for (int i = 0; i < m; ++i) if (f[i] != g[i]) return false;
for (int i = m; i < n; ++i) if (f[i] != 0) return false;
return true;
}
friend bool operator!=(const FormalPowerSeries& f, const FormalPowerSeries& g) { return not (f == g); }
/* Other Operations */
FormalPowerSeries& diff_inplace() {
const int n = size();
for (int i = 1; i < n; ++i) (*this)[i - 1] = (*this)[i] * i;
return (*this)[n - 1] = 0, *this;
}
FormalPowerSeries diff() const {
FormalPowerSeries res = *this;
res.diff_inplace();
return res;
}
FormalPowerSeries& intg_inplace() {
const int n = size();
inv_mods<value_type> invs(n);
this->resize(n + 1);
for (int i = n; i > 0; --i) (*this)[i] = (*this)[i - 1] * invs[i];
return (*this)[0] = 0, *this;
}
FormalPowerSeries intg() const {
FormalPowerSeries res = *this;
res.intg_inplace();
return res;
}
FormalPowerSeries& inv_inplace(int n = -1) { return *this = inv(n); }
// reference: https://opt-cp.com/fps-fast-algorithms/
FormalPowerSeries inv(int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).inv();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return inv_sparse(std::move(*sp_f), n);
FormalPowerSeries f_fft, g_fft;
FormalPowerSeries g{ (*this)[0].inv() };
for (int k = 1; k < n; k *= 2) {
f_fft = cut_copy(2 * k), g_fft = g.cut_copy(2 * k);
atcoder::internal::butterfly(f_fft);
atcoder::internal::butterfly(g_fft);
update_inv(k, f_fft, g_fft, g);
}
g.resize(n);
return g;
}
FormalPowerSeries& log_inplace(int n = -1) { return *this = log(n); }
FormalPowerSeries log(int n = -1) const {
assert(safe_get(0) == 1);
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).log();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return log_sparse(std::move(*sp_f), n);
FormalPowerSeries res = inv(n) * diff();
res.resize(n - 1);
return res.intg();
}
FormalPowerSeries& exp_inplace(int n = -1) { return *this = exp(n); }
// https://arxiv.org/pdf/1301.5804.pdf
FormalPowerSeries exp(int n = -1) const {
assert(safe_get(0) == 0);
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).exp();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return exp_sparse(std::move(*sp_f), n);
// h = *this
// f = exp(h) mod x ^ k
// g = f^{-1} mod x ^ k
FormalPowerSeries dh = diff();
FormalPowerSeries f{ 1 }, f_fft;
FormalPowerSeries g{ 1 }, g_fft;
for (int k = 1; k < n; k *= 2) {
f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft);
if (k > 1) update_inv(k / 2, f_fft, g_fft, g);
FormalPowerSeries t = f.cut_copy(k);
t.diff_inplace();
{
FormalPowerSeries r = dh.cut_copy(k);
r.back() = 0;
atcoder::internal::butterfly(r);
for (int i = 0; i < k; ++i) r[i] *= f_fft[i];
atcoder::internal::butterfly_inv(r);
r /= -k;
t += r;
t <<= 1, t[0] = t[k], t.pop_back();
}
t.resize(2 * k);
atcoder::internal::butterfly(t);
g_fft = g.cut_copy(2 * k);
atcoder::internal::butterfly(g_fft);
for (int i = 0; i < 2 * k; ++i) t[i] *= g_fft[i];
atcoder::internal::butterfly_inv(t);
t.resize(k);
t /= 2 * k;
FormalPowerSeries v = cut_copy(2 * k) >>= k;
t <<= k - 1;
t.intg_inplace();
for (int i = 0; i < k; ++i) v[i] -= t[k + i];
v.resize(2 * k);
atcoder::internal::butterfly(v);
for (int i = 0; i < 2 * k; ++i) v[i] *= f_fft[i];
atcoder::internal::butterfly_inv(v);
v.resize(k);
v /= 2 * k;
f.resize(2 * k);
for (int i = 0; i < k; ++i) f[k + i] = v[i];
}
f.cut(n);
return f;
}
FormalPowerSeries& pow_inplace(long long k, int n = -1) { return *this = pow(k, n); }
FormalPowerSeries pow(const long long k, int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).pow(k);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return pow_sparse(std::move(*sp_f), k, n);
if (k == 0) {
FormalPowerSeries f{ 1 };
f.resize(n);
return f;
}
int tlz = 0;
while (tlz < size() and (*this)[tlz] == 0) ++tlz;
if (tlz == size() or tlz > (n - 1) / k) return FormalPowerSeries(n, 0);
const int m = n - tlz * k;
FormalPowerSeries f = *this >> tlz;
value_type base = f[0];
return ((((f /= base).log(m) *= k).exp(m) *= base.pow(k)) <<= (tlz * k));
}
std::optional<FormalPowerSeries> safe_sqrt(int n = -1) const {
if (n < 0) n = size();
if (n < 60) return FPSNaive<mint>(cut_copy(n)).safe_sqrt();
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return safe_sqrt_sparse(std::move(*sp_f), n);
int tlz = 0;
while (tlz < size() and (*this)[tlz] == 0) ++tlz;
if (tlz == size()) return FormalPowerSeries(n, 0);
if (tlz & 1) return std::nullopt;
const int m = n - tlz / 2;
FormalPowerSeries h(this->begin() + tlz, this->end());
auto q0 = ::safe_sqrt(h[0]);
if (not q0.has_value()) return std::nullopt;
FormalPowerSeries f{ *q0 }, f_fft, g{ q0->inv() }, g_fft;
for (int k = 1; k < m; k *= 2) {
f_fft = f.cut_copy(2 * k), atcoder::internal::butterfly(f_fft);
if (k > 1) update_inv(k / 2, f_fft, g_fft, g);
g_fft = g.cut_copy(2 * k);
atcoder::internal::butterfly(g_fft);
FormalPowerSeries h_fft = h.cut_copy(2 * k);
atcoder::internal::butterfly(h_fft);
for (int i = 0; i < 2 * k; ++i) h_fft[i] = (h_fft[i] - f_fft[i] * f_fft[i]) * g_fft[i];
atcoder::internal::butterfly_inv(h_fft);
f.resize(2 * k);
const value_type iz = value_type(4 * k).inv();
for (int i = 0; i < k; ++i) f[k + i] = h_fft[k + i] * iz;
}
f.resize(m), f <<= (tlz / 2);
return f;
}
FormalPowerSeries& sqrt_inplace(int n = -1) { return *this = sqrt(n); }
FormalPowerSeries sqrt(int n = -1) const {
return *safe_sqrt(n);
}
value_type eval(value_type x) const {
value_type y = 0;
for (int i = size() - 1; i >= 0; --i) y = y * x + (*this)[i];
return y;
}
static FormalPowerSeries prod(const std::vector<FormalPowerSeries>& fs) {
if (fs.empty()) return { 1 };
std::deque<FormalPowerSeries> dq(fs.begin(), fs.end());
std::sort(dq.begin(), dq.end(), [](auto& f, auto& g) { return f.size() < g.size(); });
while (dq.size() >= 2) {
dq.push_back(dq[0] * dq[1]);
dq.pop_front();
dq.pop_front();
}
return dq.front();
}
std::optional<std::vector<std::pair<int, value_type>>> sparse_fps_format(int max_size) const {
std::vector<std::pair<int, value_type>> res;
for (int i = 0; i <= deg() and int(res.size()) <= max_size; ++i) if (value_type v = (*this)[i]; v != 0) res.emplace_back(i, v);
if (int(res.size()) > max_size) return std::nullopt;
return res;
}
private:
static void update_inv(const int k, FormalPowerSeries& f_fft, FormalPowerSeries& g_fft, FormalPowerSeries& g) {
FormalPowerSeries fg(2 * k);
for (int i = 0; i < 2 * k; ++i) fg[i] = f_fft[i] * g_fft[i];
atcoder::internal::butterfly_inv(fg);
fg >>= k, fg.resize(2 * k);
atcoder::internal::butterfly(fg);
for (int i = 0; i < 2 * k; ++i) fg[i] *= g_fft[i];
atcoder::internal::butterfly_inv(fg);
const value_type iz = value_type(2 * k).inv(), c = -iz * iz;
g.resize(2 * k);
for (int i = 0; i < k; ++i) g[k + i] = fg[i] * c;
}
static FormalPowerSeries div_fps_sparse(const FormalPowerSeries& f, const std::vector<std::pair<int, value_type>>& g, int n) {
const int siz = g.size();
assert(siz and g[0].first == 0);
const value_type inv_g0 = g[0].second.inv();
FormalPowerSeries h(n);
for (int i = 0; i < n; ++i) {
value_type v = f.safe_get(i);
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, gj] = g[idx];
if (j > i) break;
v -= gj * h[i - j];
}
h[i] = v * inv_g0;
}
return h;
}
static FormalPowerSeries inv_sparse(const std::vector<std::pair<int, value_type>>& g, const int n) {
return div_fps_sparse(FormalPowerSeries{ 1 }, g, n);
}
static FormalPowerSeries exp_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
assert(not siz or f[0].first != 0);
FormalPowerSeries g(n);
g[0] = 1;
inv_mods<value_type> invs(n);
for (int i = 1; i < n; ++i) {
value_type v = 0;
for (const auto& [j, fj] : f) {
if (j > i) break;
v += j * fj * g[i - j];
}
v *= invs[i];
g[i] = v;
}
return g;
}
static FormalPowerSeries log_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
assert(siz and f[0].first == 0 and f[0].second == 1);
FormalPowerSeries g(n);
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, fj] = f[idx];
if (j >= n) break;
g[j] = j * fj;
}
inv_mods<value_type> invs(n);
for (int i = 1; i < n; ++i) {
value_type v = g[i];
for (int idx = 1; idx < siz; ++idx) {
const auto& [j, fj] = f[idx];
if (j > i) break;
v -= fj * g[i - j] * (i - j);
}
v *= invs[i];
g[i] = v;
}
return g;
}
static FormalPowerSeries pow_sparse(const std::vector<std::pair<int, value_type>>& f, const long long k, const int n) {
if (k == 0) {
FormalPowerSeries res(n, 0);
res[0] = 1;
return res;
}
const int siz = f.size();
if (not siz) return FormalPowerSeries(n, 0);
const int p = f[0].first;
if (p > (n - 1) / k) return FormalPowerSeries(n, 0);
const value_type inv_f0 = f[0].second.inv();
const int lz = p * k;
FormalPowerSeries g(n);
g[lz] = f[0].second.pow(k);
inv_mods<value_type> invs(n);
for (int i = 1; lz + i < n; ++i) {
value_type v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g[lz + i - j] * (value_type(k) * j - (i - j));
}
v *= invs[i] * inv_f0;
g[lz + i] = v;
}
return g;
}
static std::optional<FormalPowerSeries> safe_sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
const int siz = f.size();
if (not siz) return FormalPowerSeries(n, 0);
const int p = f[0].first;
if (p % 2 == 1) return std::nullopt;
if (p / 2 >= n) return FormalPowerSeries(n, 0);
const value_type inv_f0 = f[0].second.inv();
const int lz = p / 2;
FormalPowerSeries g(n);
auto opt_g0 = ::safe_sqrt(f[0].second);
if (not opt_g0.has_value()) return std::nullopt;
g[lz] = *opt_g0;
value_type k = mint(2).inv();
inv_mods<value_type> invs(n);
for (int i = 1; lz + i < n; ++i) {
value_type v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g[lz + i - j] * (k * j - (i - j));
}
v *= invs[i] * inv_f0;
g[lz + i] = v;
}
return g;
}
static FormalPowerSeries sqrt_sparse(const std::vector<std::pair<int, value_type>>& f, const int n) {
return *safe_sqrt(f, n);
}
};
} // namespace suisen
template <typename mint>
suisen::FormalPowerSeries<mint> sqrt(suisen::FormalPowerSeries<mint> a) {
return a.sqrt();
}
template <typename mint>
suisen::FormalPowerSeries<mint> log(suisen::FormalPowerSeries<mint> a) {
return a.log();
}
template <typename mint>
suisen::FormalPowerSeries<mint> exp(suisen::FormalPowerSeries<mint> a) {
return a.exp();
}
template <typename mint, typename T>
suisen::FormalPowerSeries<mint> pow(suisen::FormalPowerSeries<mint> a, T b) {
return a.pow(b);
}
template <typename mint>
suisen::FormalPowerSeries<mint> inv(suisen::FormalPowerSeries<mint> a) {
return a.inv();
}
#include <utility>
namespace suisen {
template <typename FPSType>
struct RationalFPS {
using mint = typename FPSType::value_type;
FPSType num, den;
RationalFPS(const FPSType& num = { 0 }, const FPSType& den = { 1 }) : num(num), den(den) {}
RationalFPS(const std::pair<FPSType, FPSType>& p) : num(p.first), den(p.second) {}
FPSType to_fps(int n) const {
int dlz = 0;
while (dlz < den.size() and den[dlz] == 0) ++dlz;
int nlz = 0;
while (nlz < num.size() and num[nlz] == 0) ++nlz;
assert(dlz != den.size());
if (nlz == num.size()) {
return FPSType(n, mint(0));
}
assert(dlz <= nlz);
return ((num >> dlz) * (den >> dlz).inv(n)).cut(n);
}
RationalFPS<FPSType> operator+() const { return *this; }
RationalFPS<FPSType> operator-() const { return { -num, den }; }
friend RationalFPS<FPSType> operator+(const RationalFPS& lhs, const RationalFPS& rhs) {
return { lhs.num * rhs.den + lhs.den * rhs.num, lhs.den * rhs.den };
}
friend RationalFPS<FPSType> operator-(const RationalFPS& lhs, const RationalFPS& rhs) {
return { lhs.num * rhs.den - lhs.den * rhs.num, lhs.den * rhs.den };
}
friend RationalFPS<FPSType> operator*(const RationalFPS& lhs, const RationalFPS& rhs) {
return { lhs.num * rhs.num, lhs.den * rhs.den };
}
friend RationalFPS<FPSType> operator*(const RationalFPS& lhs, const mint& val) {
return { lhs.num * val, lhs.den };
}
friend RationalFPS<FPSType> operator/(const RationalFPS& lhs, const mint& val) {
return { lhs.num, lhs.den * val };
}
friend RationalFPS<FPSType> operator*(const mint& val, const RationalFPS& lhs) {
return { lhs.num * val, lhs.den };
}
friend RationalFPS<FPSType> operator/(const mint& val, const RationalFPS& lhs) {
return { lhs.den * val, lhs.num };
}
RationalFPS<FPSType>& operator+=(const RationalFPS& rhs) { return *this = *this + rhs; }
RationalFPS<FPSType>& operator-=(const RationalFPS& rhs) { return *this = *this - rhs; }
RationalFPS<FPSType>& operator*=(const RationalFPS& rhs) { return *this = *this * rhs; }
RationalFPS<FPSType>& operator*=(const mint& val) { return num *= val, *this; }
RationalFPS<FPSType>& operator/=(const mint& val) { return den *= val, *this; }
RationalFPS<FPSType> inv() const { return { den, num }; }
RationalFPS<FPSType>& inv_inplace() { return std::swap(num, den), * this; }
FPSType normalize() {
auto [q, r] = num.div_mod(den);
num = std::move(r);
return q;
}
static RationalFPS<FPSType> sum(const std::vector<RationalFPS<FPSType>>& fs) {
auto comp = [](const RationalFPS<FPSType>& f, const RationalFPS<FPSType>& g) {
return f.den.size() > g.den.size();
};
std::priority_queue<RationalFPS<FPSType>, std::vector<RationalFPS<FPSType>>, decltype(comp)> pq{ comp };
for (const auto& f : fs) pq.push(f);
while (pq.size() > 1) {
auto f = pq.top();
pq.pop();
auto g = pq.top();
pq.pop();
pq.emplace(f + g);
}
return pq.top();
}
static RationalFPS<FPSType> prod(const std::vector<RationalFPS<FPSType>>& fs) {
std::vector<FPSType> nums, dens;
for (const auto &f : fs) nums.push_back(f.num), dens.push_back(f.den);
return { FPSType::prod(nums), FPSType::prod(dens) };
}
};
} // namespace suisen
using namespace suisen;
using FPS = FormalPowerSeries<mint>;
using RFPS = RationalFPS<FPS>;
int main() {
int n, m;
std::cin >> n >> m;
std::vector<mint> a(n);
for (auto &e : a) std::cin >> e;
std::vector<RFPS> fs(n);
for (int i = 0; i < n; ++i) {
fs[i] = { FPS{1}, FPS{1, -a[i]} };
}
FPS f = RFPS::sum(fs).to_fps(m + 1);
for (int i = 1; i <= m; ++i) {
std::cout << f[i] << " \n"[i == m];
}
return 0;
}