結果
| 問題 | No.2005 Sum of Power Sums |
| ユーザー |
 suisen
|
| 提出日時 | 2022-07-10 22:34:42 |
| 言語 | C++17 (gcc 12.3.0 + boost 1.83.0) |
| 結果 |
AC
|
| 実行時間 | 1,287 ms / 2,000 ms |
| コード長 | 47,062 bytes |
| コンパイル時間 | 3,206 ms |
| コンパイル使用メモリ | 153,388 KB |
| 実行使用メモリ | 7,108 KB |
| 最終ジャッジ日時 | 2024-07-02 07:28:54 |
| 合計ジャッジ時間 | 28,552 ms |
|
ジャッジサーバーID （参考情報） |
judge5 / 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 | 995 ms
6,812 KB |
| testcase_01 | AC | 1,001 ms
6,944 KB |
| testcase_02 | AC | 1,257 ms
6,940 KB |
| testcase_03 | AC | 993 ms
6,940 KB |
| testcase_04 | AC | 991 ms
6,944 KB |
| testcase_05 | AC | 995 ms
6,940 KB |
| testcase_06 | AC | 989 ms
6,940 KB |
| testcase_07 | AC | 992 ms
6,940 KB |
| testcase_08 | AC | 988 ms
6,940 KB |
| testcase_09 | AC | 985 ms
6,940 KB |
| testcase_10 | AC | 1,074 ms
6,944 KB |
| testcase_11 | AC | 1,116 ms
6,940 KB |
| testcase_12 | AC | 1,267 ms
6,940 KB |
| testcase_13 | AC | 1,270 ms
6,940 KB |
| testcase_14 | AC | 1,019 ms
7,100 KB |
| testcase_15 | AC | 1,286 ms
6,972 KB |
| testcase_16 | AC | 1,287 ms
7,108 KB |
| testcase_17 | AC | 1,278 ms
6,940 KB |
| testcase_18 | AC | 1,156 ms
6,940 KB |
| testcase_19 | AC | 1,156 ms
6,940 KB |
| testcase_20 | AC | 1,175 ms
7,108 KB |
ソースコード
#define PROBLEM "https://yukicoder.me/problems/no/2005"
#include <iostream>
#include <atcoder/convolution>
#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 <algorithm>
#include <cassert>
#include <iostream>
#include <cmath>
#include <limits>
#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
#include <optional>
/**
* 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> optional_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 *optional_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{};
}
namespace suisen {
template <typename T>
struct FPSNaive : std::vector<T> {
static inline int MAX_DEG = 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) {}
static void set_max_deg(int max_deg) {
FPSNaive<T>::MAX_DEG = max_deg;
}
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 max_deg) {
if (deg() > max_deg) this->resize(std::max(0, max_deg + 1));
return *this;
}
FPSNaive cut(int max_deg) const {
return FPSNaive(*this).cut_inplace(max_deg);
}
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_DEG + 1, n + m - 1));
for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) {
if (i + j > MAX_DEG) 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, FPSNaive g) { return std::move(div_mod(std::move(f), std::move(g)).first); }
friend FPSNaive operator%(FPSNaive f, FPSNaive g) { return std::move(div_mod(std::move(f), std::move(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; }
friend std::pair<FPSNaive, FPSNaive> div_mod(FPSNaive f, FPSNaive g) {
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 - 1) };
}
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 max_deg) const {
if (this->empty() or g.empty()) return FPSNaive{};
const int n = size(), m = g.size();
FPSNaive h(std::min(max_deg + 1, n + m - 1));
for (int i = 0; i < n; ++i) for (int j = 0; j < m; ++j) {
if (i + j > max_deg) break;
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_DEG) g.cut_inplace(MAX_DEG);
return g;
}
FPSNaive inv(int max_deg) const {
FPSNaive g(max_deg + 1);
const value_type inv_f0 = ::inv(unsafe_get(0));
g.unsafe_get(0) = inv_f0;
for (int i = 1; i <= max_deg; ++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 max_deg) const {
assert(unsafe_get(0) == value_type{ 0 });
FPSNaive g(max_deg + 1);
g.unsafe_get(0) = value_type{ 1 };
for (int i = 1; i <= max_deg; ++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 max_deg) const {
assert(unsafe_get(0) == value_type{ 1 });
FPSNaive g(max_deg + 1);
g.unsafe_get(0) = value_type{ 0 };
for (int i = 1; i <= max_deg; ++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 max_deg) const {
if (k == 0) return { value_type{ 1 } };
int z = 0;
while (z < size() and unsafe_get(z) == value_type{ 0 }) ++z;
if (z == size() or z > max_deg / k) return FPSNaive{};
const int d = max_deg - z * k;
FPSNaive g(d + 1);
const value_type inv_f0 = ::inv(unsafe_get(z));
g.unsafe_get(0) = unsafe_get(z).pow(k);
for (int i = 1; i <= d; ++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> optional_sqrt(int max_deg) const {
int dl = 0;
while (dl < size() and unsafe_get(dl) == value_type{ 0 }) ++dl;
if (dl == size()) return FPSNaive{};
if (dl & 1) return std::nullopt;
const int d = max_deg - dl / 2;
FPSNaive g(d + 1);
auto opt_g0 = ::optional_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 <= d; ++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 max_deg) const {
return *optional_sqrt(max_deg);
}
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>
auto sqrt(suisen::FPSNaive<mint> a) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
return a.sqrt(suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}
template <typename mint>
auto log(suisen::FPSNaive<mint> a) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
return a.log(suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}
template <typename mint>
auto exp(suisen::FPSNaive<mint> a) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
return a.exp(suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}
template <typename mint, typename T>
auto pow(suisen::FPSNaive<mint> a, T b) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
return a.pow(b, suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}
template <typename mint>
auto inv(suisen::FPSNaive<mint> a) -> decltype(mint::mod(), suisen::FPSNaive<mint>{}) {
return a.inv(suisen::FPSNaive<mint>::MAX_DEG == std::numeric_limits<int>::max() / 2 ? suisen::FPSNaive<mint>::MAX_DEG : a.deg());
}
namespace suisen {
template <typename mint>
using convolution_t = std::vector<mint>(*)(const std::vector<mint>&, const std::vector<mint>&);
template <typename mint>
struct FPS : public std::vector<mint> {
using std::vector<mint>::vector;
FPS(const std::initializer_list<mint> l) : std::vector<mint>::vector(l) {}
FPS(const std::vector<mint>& v) : std::vector<mint>::vector(v) {}
FPS(std::vector<mint>&& v) : std::vector<mint>::vector(std::move(v)) {}
static void set_multiplication(convolution_t<mint> multiplication) {
FPS<mint>::mult = multiplication;
}
const mint operator[](int n) const noexcept { return n <= deg() ? unsafe_get(n) : 0; }
mint& operator[](int n) noexcept { ensure_deg(n); return unsafe_get(n); }
int size() const noexcept { return std::vector<mint>::size(); }
int deg() const noexcept { return size() - 1; }
int normalize() {
while (this->size() and this->back() == 0) this->pop_back();
return deg();
}
FPS& pre_inplace(int max_deg) noexcept {
if (deg() > max_deg) this->resize(std::max(0, max_deg + 1));
return *this;
}
FPS pre(int max_deg) const noexcept { return FPS(*this).pre_inplace(max_deg); }
FPS operator+() const { return FPS(*this); }
FPS operator-() const {
FPS f(*this);
for (auto& e : f) e = mint::mod() - e;
return f;
}
FPS& operator++() { ++(*this)[0]; return *this; }
FPS& operator--() { --(*this)[0]; return *this; }
FPS& operator+=(const mint x) { (*this)[0] += x; return *this; }
FPS& operator-=(const mint x) { (*this)[0] -= x; return *this; }
FPS& operator+=(const FPS& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) += g.unsafe_get(i);
return *this;
}
FPS& operator-=(const FPS& g) {
ensure_deg(g.deg());
for (int i = 0; i <= g.deg(); ++i) unsafe_get(i) -= g.unsafe_get(i);
return *this;
}
FPS& operator*=(const FPS& g) { return *this = FPS<mint>::mult(*this, g); }
FPS& operator*=(const mint x) {
for (auto& e : *this) e *= x;
return *this;
}
FPS& operator/=(FPS g) {
const int fd = normalize(), gd = g.normalize();
assert(gd >= 0);
if (fd < gd) { this->clear(); return *this; }
if (gd == 0) return *this *= g.unsafe_get(0).inv();
static constexpr int THRESHOLD_NAIVE_POLY_QUOTIENT = 256;
if (gd <= THRESHOLD_NAIVE_POLY_QUOTIENT) {
*this = std::move(naive_div_inplace(std::move(g), gd).first);
return *this;
}
std::reverse(this->begin(), this->end()), std::reverse(g.begin(), g.end());
const int k = fd - gd;
*this *= g.inv_inplace(k), this->resize(k + 1);
std::reverse(this->begin(), this->end());
return *this;
}
FPS& operator%=(FPS g) {
int fd = normalize(), gd = g.normalize();
assert(gd >= 0);
if (fd < gd) return *this;
if (gd == 0) { this->clear(); return *this; }
static constexpr int THRESHOLD_NAIVE_REMAINDER = 256;
if (gd <= THRESHOLD_NAIVE_REMAINDER) return naive_div_inplace(std::move(g), gd).second;
*this -= g * (*this / g);
return pre_inplace(gd - 1);
}
FPS& operator<<=(const int shamt) {
this->insert(this->begin(), shamt, 0);
return *this;
}
FPS& operator>>=(const int shamt) {
if (shamt > size()) this->clear();
else this->erase(this->begin(), this->begin() + shamt);
return *this;
}
friend FPS operator+(FPS f, const FPS& g) { f += g; return f; }
friend FPS operator+(FPS f, const mint x) { f += x; return f; }
friend FPS operator-(FPS f, const FPS& g) { f -= g; return f; }
friend FPS operator-(FPS f, const mint x) { f -= x; return f; }
friend FPS operator*(FPS f, const FPS& g) { f *= g; return f; }
friend FPS operator*(FPS f, const mint x) { f *= x; return f; }
friend FPS operator/(FPS f, const FPS& g) { f /= g; return f; }
friend FPS operator%(FPS f, const FPS& g) { f %= g; return f; }
friend FPS operator*(const mint x, FPS f) { f *= x; return f; }
friend FPS operator<<(FPS f, const int shamt) { f <<= shamt; return f; }
friend FPS operator>>(FPS f, const int shamt) { f >>= shamt; return f; }
friend bool operator==(const FPS& f, const FPS& g) {
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;
}
FPS& diff_inplace() {
if (this->size() == 0) return *this;
for (int i = 1; i <= deg(); ++i) unsafe_get(i - 1) = unsafe_get(i) * i;
this->pop_back();
return *this;
}
FPS& intg_inplace() {
int d = deg();
ensure_deg(d + 1);
for (int i = d; i >= 0; --i) unsafe_get(i + 1) = unsafe_get(i) * invs[i + 1];
unsafe_get(0) = 0;
return *this;
}
FPS& inv_inplace(const int max_deg) {
if (max_deg <= 60) return *this = FPSNaive<mint>(this->begin(), this->end()).inv(max_deg);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return *this = inv_sparse(std::move(*sp_f), max_deg);
FPS res{ unsafe_get(0).inv() };
for (int k = 1; k <= max_deg; k *= 2) {
FPS tmp(this->pre(k * 2) * (res * res));
res *= 2, res -= tmp.pre_inplace(2 * k);
}
return *this = std::move(res), pre_inplace(max_deg);
}
FPS& log_inplace(const int max_deg) {
if (max_deg <= 60) return *this = FPSNaive<mint>(this->begin(), this->end()).log(max_deg);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return *this = log_sparse(std::move(*sp_f), max_deg);
FPS f_inv = inv(max_deg);
diff_inplace(), *this *= f_inv, pre_inplace(max_deg - 1), intg_inplace();
return *this;
}
FPS& exp_inplace(const int max_deg) {
if (max_deg <= 60) return *this = FPSNaive<mint>(this->begin(), this->end()).exp(max_deg);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return *this = exp_sparse(std::move(*sp_f), max_deg);
FPS res{ 1 };
for (int k = 1; k <= max_deg; k *= 2) res *= ++(pre(k * 2) - res.log(k * 2)), res.pre_inplace(k * 2);
return *this = std::move(res), pre_inplace(max_deg);
}
FPS& sqrt_inplace(const int max_deg) {
return *this = sqrt(max_deg);
}
FPS& pow_inplace(const long long k, const int max_deg) {
if (k == 0) return *this = { mint{ 1 } };
if (max_deg <= 60) return *this = FPSNaive<mint>(this->begin(), this->end()).pow(k, max_deg);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return *this = pow_sparse(std::move(*sp_f), k, max_deg);
int tlz = 0;
while (tlz <= deg() and unsafe_get(tlz) == 0) ++tlz;
if (tlz > deg() or tlz > max_deg / k) return this->clear(), *this;
const int d = max_deg - tlz * k;
*this >>= tlz;
mint base = (*this)[0];
*this *= base.inv(), log_inplace(d), *this *= k, exp_inplace(d), *this *= base.pow(k);
return *this <<= tlz * k;
}
FPS diff() const { FPS f{ *this }; f.diff_inplace(); return f; }
FPS intg() const { FPS f{ *this }; f.intg_inplace(); return f; }
FPS inv(const int max_deg) const { FPS f{ *this }; f.inv_inplace(max_deg); return f; }
FPS log(const int max_deg) const { FPS f{ *this }; f.log_inplace(max_deg); return f; }
FPS exp(const int max_deg) const { FPS f{ *this }; f.exp_inplace(max_deg); return f; }
std::optional<FPS> optional_sqrt(const int max_deg) {
if (max_deg <= 60) return FPSNaive<mint>(this->begin(), this->end()).optional_sqrt(max_deg);
if (auto sp_f = sparse_fps_format(15); sp_f.has_value()) return optional_sqrt_sparse(std::move(*sp_f), max_deg);
int tlz = 0;
while (tlz <= deg() and unsafe_get(tlz) == 0) ++tlz;
if (tlz > deg()) return FPS{};
if (tlz % 2 == 1) return std::nullopt;
int max_deg2 = max_deg - tlz / 2;
FPS f(this->begin() + tlz, this->end());
auto opt_res0 = ::optional_sqrt(f[0]);
if (not opt_res0.has_value()) return std::nullopt;
FPS res{ *opt_res0 };
mint inv_2 = mint(2).inv();
for (int k = 1; k <= max_deg2; k *= 2) {
res = ((f.pre(k * 2) * res.inv(2 * k)).pre_inplace(2 * k) += res) *= inv_2;
}
return *this = std::move(res <<= tlz / 2), pre_inplace(max_deg);
}
FPS sqrt(const int max_deg) const { return *optional_sqrt(max_deg); }
FPS pow(const long long k, const int max_deg) const { FPS f{ *this }; f.pow_inplace(k, max_deg); return f; }
mint eval(mint x) const {
mint y = 0;
for (int i = size() - 1; i >= 0; --i) y = y * x + unsafe_get(i);
return y;
}
private:
static inline inv_mods<mint> invs;
static convolution_t<mint> mult;
void ensure_deg(int d) { if (deg() < d) this->resize(d + 1, 0); }
const mint& unsafe_get(int i) const { return std::vector<mint>::operator[](i); }
mint& unsafe_get(int i) { return std::vector<mint>::operator[](i); }
std::optional<std::vector<std::pair<int, mint>>> sparse_fps_format(int max_size) const {
std::vector<std::pair<int, mint>> res;
for (int i = 0; i <= deg() and int(res.size()) <= max_size; ++i) if (mint v = unsafe_get(i); v != 0) res.emplace_back(i, v);
if (int(res.size()) > max_size) return std::nullopt;
return res;
}
std::pair<FPS, FPS&> naive_div_inplace(FPS&& g, const int gd) {
const int k = deg() - gd;
mint head_inv = g.unsafe_get(gd).inv();
FPS q(k + 1);
for (int i = k; i >= 0; --i) {
mint div = this->unsafe_get(i + gd) * head_inv;
q.unsafe_get(i) = div;
for (int j = 0; j <= gd; ++j) this->unsafe_get(i + j) -= div * g.unsafe_get(j);
}
return { q, pre_inplace(gd - 1) };
}
static FPS div_fps_sparse(const FPS &f, const std::vector<std::pair<int, mint>> &g, const int max_deg) {
const int siz = g.size();
assert(siz and g[0].first == 0);
const mint inv_g0 = g[0].second.inv();
FPS h(max_deg + 1);
for (int i = 0; i <= max_deg; ++i) {
mint v = f[i];
for (int idx = 1; idx < siz; ++idx) {
const auto &[j, gj] = g[idx];
if (j > i) break;
v -= gj * h.unsafe_get(i - j);
}
h.unsafe_get(i) = v * inv_g0;
}
return h;
}
static FPS inv_sparse(const std::vector<std::pair<int, mint>> &g, const int max_deg) {
return div_fps_sparse(FPS { mint{1} }, g, max_deg);
}
static FPS exp_sparse(const std::vector<std::pair<int, mint>> &f, const int max_deg) {
const int siz = f.size();
assert(not siz or f[0].first != 0);
FPS g(max_deg + 1);
g[0] = 1;
for (int i = 1; i <= max_deg; ++i) {
mint v = 0;
for (const auto &[j, fj] : f) {
if (j > i) break;
v += j * fj * g.unsafe_get(i - j);
}
v *= invs[i];
g.unsafe_get(i) = v;
}
return g;
}
static FPS log_sparse(const std::vector<std::pair<int, mint>> &f, const int max_deg) {
const int siz = f.size();
assert(siz and f[0].first == 0 and f[0].second == 1);
FPS g(max_deg + 1);
for (int idx = 1; idx < siz; ++idx) {
const auto &[j, fj] = f[idx];
if (j > max_deg) break;
g.unsafe_get(j) = j * fj;
}
for (int i = 1; i <= max_deg; ++i) {
mint v = g.unsafe_get(i);
for (int idx = 1; idx < siz; ++idx) {
const auto &[j, fj] = f[idx];
if (j > i) break;
v -= fj * g.unsafe_get(i - j) * (i - j);
}
v *= invs[i];
g.unsafe_get(i) = v;
}
return g;
}
static FPS pow_sparse(const std::vector<std::pair<int, mint>> &f, const long long k, const int max_deg) {
if (k == 0) return FPS { mint{1} };
const int siz = f.size();
if (not siz) return FPS{};
const int p = f[0].first;
if (p >= max_deg / k + 1) return FPS{};
const mint inv_f0 = f[0].second.inv();
const int lz = p * k;
FPS g(max_deg + 1);
g[lz] = f[0].second.pow(k);
for (int i = 1; lz + i <= max_deg; ++i) {
mint v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g.unsafe_get(lz + i - j) * (mint(k) * j - (i - j));
}
v *= invs[i] * inv_f0;
g.unsafe_get(lz + i) = v;
}
return g;
}
static std::optional<FPS> optional_sqrt_sparse(const std::vector<std::pair<int, mint>> &f, const int max_deg) {
const int siz = f.size();
if (not siz) return FPS{};
const int p = f[0].first;
if (p % 2 == 1) return std::nullopt;
if (p / 2 > max_deg) return FPS{};
const mint inv_f0 = f[0].second.inv();
const int lz = p / 2;
FPS g(max_deg + 1);
auto opt_g0 = ::optional_sqrt(f[0].second);
if (not opt_g0.has_value()) return std::nullopt;
g[lz] = *opt_g0;
mint k = mint(2).inv();
for (int i = 1; lz + i <= max_deg; ++i) {
mint v = 0;
for (int idx = 1; idx < siz; ++idx) {
auto [j, fj] = f[idx];
j -= p;
if (j > i) break;
v += fj * g.unsafe_get(lz + i - j) * (k * j - (i - j));
}
v *= invs[i] * inv_f0;
g.unsafe_get(lz + i) = v;
}
return g;
}
static FPS sqrt_sparse(const std::vector<std::pair<int, mint>> &f, const int max_deg) {
return *optional_sqrt(f, max_deg);
}
};
template <typename mint>
convolution_t<mint> FPS<mint>::mult = [](const auto&, const auto&) {
std::cerr << "convolution function is not available." << std::endl;
assert(false);
return std::vector<mint>{};
};
} // namespace suisen
template <typename mint>
auto sqrt(suisen::FPS<mint> a) -> decltype(mint::mod(), suisen::FPS<mint>{}) {
return a.sqrt(a.deg());
}
template <typename mint>
auto log(suisen::FPS<mint> a) -> decltype(mint::mod(), suisen::FPS<mint>{}) {
return a.log(a.deg());
}
template <typename mint>
auto exp(suisen::FPS<mint> a) -> decltype(mint::mod(), suisen::FPS<mint>{}) {
return a.exp(a.deg());
}
template <typename mint, typename T>
auto pow(suisen::FPS<mint> a, T b) -> decltype(mint::mod(), suisen::FPS<mint>{}) {
return a.pow(b, a.deg());
}
template <typename mint>
auto inv(suisen::FPS<mint> a) -> decltype(mint::mod(), suisen::FPS<mint>{}) {
return a.inv(a.deg());
}
namespace suisen {
template <typename T, typename U = T>
struct factorial {
factorial() {}
factorial(int n) { ensure(n); }
static void ensure(const int n) {
int sz = _fac.size();
if (n + 1 <= sz) return;
int new_size = std::max(n + 1, sz * 2);
_fac.resize(new_size), _fac_inv.resize(new_size);
for (int i = sz; i < new_size; ++i) _fac[i] = _fac[i - 1] * i;
_fac_inv[new_size - 1] = U(1) / _fac[new_size - 1];
for (int i = new_size - 1; i > sz; --i) _fac_inv[i - 1] = _fac_inv[i] * i;
}
T fac(const int i) {
ensure(i);
return _fac[i];
}
T operator()(int i) {
return fac(i);
}
U fac_inv(const int i) {
ensure(i);
return _fac_inv[i];
}
U binom(const int n, const int r) {
if (n < 0 or r < 0 or n < r) return 0;
ensure(n);
return _fac[n] * _fac_inv[r] * _fac_inv[n - r];
}
U perm(const int n, const int r) {
if (n < 0 or r < 0 or n < r) return 0;
ensure(n);
return _fac[n] * _fac_inv[n - r];
}
private:
static std::vector<T> _fac;
static std::vector<U> _fac_inv;
};
template <typename T, typename U>
std::vector<T> factorial<T, U>::_fac{ 1 };
template <typename T, typename U>
std::vector<U> factorial<T, U>::_fac_inv{ 1 };
} // namespace suisen
#include <cstdint>
#include <numeric>
namespace suisen {
// referece: https://37zigen.com/linear-sieve/
class LinearSieve {
public:
LinearSieve(const int n) : _n(n), min_prime_factor(std::vector<int>(n + 1)) {
std::iota(min_prime_factor.begin(), min_prime_factor.end(), 0);
prime_list.reserve(_n / 20);
for (int d = 2; d <= _n; ++d) {
if (min_prime_factor[d] == d) prime_list.push_back(d);
const int prime_max = std::min(min_prime_factor[d], _n / d);
for (int prime : prime_list) {
if (prime > prime_max) break;
min_prime_factor[prime * d] = prime;
}
}
}
int prime_num() const noexcept { return prime_list.size(); }
/**
* Returns a vector of primes in [0, n].
* It is guaranteed that the returned vector is sorted in ascending order.
*/
const std::vector<int>& get_prime_list() const noexcept {
return prime_list;
}
const std::vector<int>& get_min_prime_factor() const noexcept { return min_prime_factor; }
/**
* Returns a vector of `{ prime, index }`.
* It is guaranteed that the returned vector is sorted in ascending order.
*/
std::vector<std::pair<int, int>> factorize(int n) const noexcept {
assert(0 < n and n <= _n);
std::vector<std::pair<int, int>> prime_powers;
while (n > 1) {
int p = min_prime_factor[n], c = 0;
do { n /= p, ++c; } while (n % p == 0);
prime_powers.emplace_back(p, c);
}
return prime_powers;
}
private:
const int _n;
std::vector<int> min_prime_factor;
std::vector<int> prime_list;
};
} // namespace suisen
namespace suisen {
// returns { 0^k, 1^k, ..., n^k }
template <typename mint>
std::vector<mint> powers(uint32_t n, uint64_t k) {
const auto mpf = LinearSieve(n).get_min_prime_factor();
std::vector<mint> res(n + 1);
res[0] = k == 0;
for (uint32_t i = 1; i <= n; ++i) res[i] = i == 1 ? 1 : uint32_t(mpf[i]) == i ? mint(i).pow(k) : res[mpf[i]] * res[i / mpf[i]];
return res;
}
} // namespace suisen
// reference: https://en.wikipedia.org/wiki/Eulerian_number
namespace suisen {
template <typename mint>
std::vector<mint> eulerian_number(uint32_t n) {
if (n == 0) return {};
factorial<mint> fac(n + 1);
const uint32_t h = (n + 1) >> 1;
FPS<mint> f = powers<mint>(h, n);
f.erase(f.begin());
FPS<mint> g(h);
for (uint32_t i = 0; i < h; ++i) {
mint v = fac.binom(n + 1, i);
g[i] = i & 1 ? -v : v;
}
FPS<mint> res = f * g;
res.resize(n);
for (uint32_t i = h; i < n; ++i) res[i] = res[n - 1 - i];
return res;
}
template <typename mint>
std::vector<std::vector<mint>> eulerian_number_table(uint32_t n) {
if (n == 0) return {};
std::vector dp(n + 1, std::vector<mint>{});
for (uint32_t i = 1; i <= n; ++i) {
dp[i].resize(i);
dp[i][0] = dp[i][i - 1] = 1;
for (uint32_t j = 1; j < i - 1; ++j) dp[i][j] = (i - j) * dp[i - 1][j - 1] + (j + 1) * dp[i - 1][j];
}
return dp;
}
} // namespace suisen
/**
* [Idea] reference : https://motsu-xe.hatenablog.com/entry/2021/05/13/224016
*
* SWAG + simulate a deque with 2 stacks
*
* [Operations] reference : https://www.slideshare.net/catupper/amortize-analysis-of-deque-with-2-stack
*
* `l`, `r` is a stack of { value, sum }
*
* accumulate
* <---------- ------> fold values from inside
* ( l ][ r )
*
* pop_front:
* 1. `l` is not empty
* ( l ][ r ) -> ( l ][ r ) # pop from `l`. O(1)
* 2. `l` is empty
* (][ r ) -> ( l ][ r ) # split `r` at its middle point. amortized O(1)
* ( l ][ r ) -> ( l ][ r ) # pop from `l`. O(1)
*
* pop_back:
* 1. `r` is not empty
* ( l ][ r ) -> ( l ][ r ) # pop from `r`. O(1)
* 2. `r` is empty
* ( l ][) -> ( l ][ r ) # split `l` at its middle point. amortized O(1)
* ( l ][ r ) -> ( l ][ r ) # pop from `r`. O(1)
*
* push_front:
* ( l ][ r ) -> ( l ][ r ) # push to `l`. O(1)
*
* push_back:
* ( l ][ r ) -> ( l ][ r ) # push to `r`. O(1)
*/
namespace suisen {
template <typename T, T(*op)(T, T), T(*e)()>
struct DequeAggregation {
struct DequeAggregationIterator {
using difference_type = int;
using value_type = T;
using pointer = value_type*;
using reference = value_type&;
using iterator_category = std::random_access_iterator_tag;
using fi_iterator_type = typename std::vector<std::pair<value_type, value_type>>::const_reverse_iterator;
using se_iterator_type = typename std::vector<std::pair<value_type, value_type>>::const_iterator;
fi_iterator_type it_l;
fi_iterator_type it_l_end;
se_iterator_type it_r_begin;
se_iterator_type it_r;
DequeAggregationIterator& operator++() {
if (it_l == it_l_end) ++it_r;
else ++it_l;
return *this;
}
DequeAggregationIterator operator++(int) { DequeAggregationIterator ret = *this; ++(*this); return ret; }
DequeAggregationIterator& operator--() {
if (it_r == it_r_begin) --it_l;
else --it_r;
return *this;
}
DequeAggregationIterator operator--(int) { DequeAggregationIterator ret = *this; --(*this); return ret; }
DequeAggregationIterator& operator+=(difference_type dif) {
if (dif < 0) return *this -= -dif;
if (int d = it_l_end - it_l; d < dif) it_l = it_l_end, it_r += dif - d;
else it_l += dif;
return *this;
}
DequeAggregationIterator operator+(difference_type dif) const { DequeAggregationIterator it = *this; it += dif; return it; }
DequeAggregationIterator& operator-=(difference_type dif) {
if (dif < 0) return *this += -dif;
if (int d = it_r - it_r_begin; d < dif) it_r = it_r_begin, it_l -= dif - d;
else it_r -= dif;
return *this;
}
DequeAggregationIterator operator-(difference_type dif) const { DequeAggregationIterator it = *this; it -= dif; return it; }
difference_type operator-(const DequeAggregationIterator &rhs) const {
difference_type d1 = it_l == it_l_end ? it_r - it_r_begin : it_l - it_l_end;
difference_type d2 = rhs.it_l == rhs.it_l_end ? rhs.it_r - rhs.it_r_begin : rhs.it_l - rhs.it_l_end;
return d1 - d2;
}
const value_type& operator[](difference_type i) const { return *((*this) + i); }
const value_type& operator*() const { return it_l == it_l_end ? it_r->first : it_l->first; }
bool operator!=(const DequeAggregationIterator &rhs) const { return it_l != rhs.it_l or it_r != rhs.it_r; }
bool operator==(const DequeAggregationIterator &rhs) const { return not (*this != rhs); }
bool operator<(const DequeAggregationIterator &rhs) const { return (*this) - rhs < 0; }
bool operator<=(const DequeAggregationIterator &rhs) const { return (*this) - rhs <= 0; }
bool operator>(const DequeAggregationIterator &rhs) const { return (*this) - rhs > 0; }
bool operator>=(const DequeAggregationIterator &rhs) const { return (*this) - rhs >= 0; }
};
using iterator = DequeAggregationIterator;
using difference_type = typename iterator::difference_type;
using value_type = typename iterator::value_type;
using pointer = typename iterator::pointer;
using reference = typename iterator::reference;
DequeAggregation() = default;
template <typename InputIterator, std::enable_if_t<std::is_constructible_v<value_type, typename InputIterator::value_type>, std::nullptr_t> = nullptr>
DequeAggregation(InputIterator first, InputIterator last) {
for (; first != last; ++first) push_back(*first);
}
template <typename Container, std::enable_if_t<std::is_constructible_v<value_type, typename Container::value_type>, std::nullptr_t> = nullptr>
DequeAggregation(const Container &c) : DequeAggregation(std::begin(c), std::end(c)) {}
value_type prod() const {
return op(prod(_st_l), prod(_st_r));
}
void push_back(const value_type &val) { _st_r.emplace_back(val, op(prod(_st_r), val)); }
void push_front(const value_type &val) { _st_l.emplace_back(val, op(val, prod(_st_l))); }
void pop_back() {
if (_st_r.size()) return _st_r.pop_back();
const int siz = _st_l.size();
const int l = siz >> 1, r = siz - l;
assert(r); // <=> siz > 0
for (int i = r - 1; i > 0; --i) push_back(std::move(_st_l[i].first));
_st_l.erase(_st_l.begin(), _st_l.begin() + r);
if (l == 0) return;
_st_l[0].second = _st_l[0].first;
for (int i = 1; i < l; ++i) _st_l[i].second = op(_st_l[i].first, _st_l[i - 1].second);
}
void pop_front() {
if (_st_l.size()) return _st_l.pop_back();
const int siz = _st_r.size();
const int r = siz >> 1, l = siz - r;
assert(l); // <=> siz > 0
for (int i = l - 1; i > 0; --i) push_front(std::move(_st_r[i].first));
_st_r.erase(_st_r.begin(), _st_r.begin() + l);
if (r == 0) return;
_st_r[0].second = _st_r[0].first;
for (int i = 1; i < r; ++i) _st_r[i].second = op(_st_r[i - 1].second, _st_r[i].first);
}
const value_type& front() const { return _st_l.size() ? _st_l.back().first : _st_r.front().first; }
const value_type& back() const { return _st_r.size() ? _st_r.back().first : _st_l.front().first; }
const value_type& operator[](int i) const {
const int k = i - _st_l.size();
return k < 0 ? _st_l[~k].first : _st_r[k].first;
}
int size() const { return _st_l.size() + _st_r.size(); }
void clear() { _st_l.clear(), _st_r.clear(); }
void shrink_to_fit() { _st_l.shrink_to_fit(), _st_r.shrink_to_fit(); }
iterator begin() const { return iterator { _st_l.rbegin(), _st_l.rend(), _st_r.begin(), _st_r.begin() }; }
iterator end() const { return iterator { _st_l.rend(), _st_l.rend(), _st_r.begin(), _st_r.end() }; }
iterator cbegin() const { return begin(); }
iterator cend() const { return end(); }
private:
std::vector<std::pair<value_type, value_type>> _st_l, _st_r;
value_type prod(const std::vector<std::pair<value_type, value_type>> &st) const {
return st.empty() ? e() : st.back().second;
}
};
} // namespace suisen
mint op(mint x, mint y) {
return x * y;
}
mint e() {
return 1;
}
constexpr uint32_t K_MAX = 5000;
int main() {
suisen::FPS<mint>::set_multiplication([](const auto &a, const auto &b) { return atcoder::convolution(a, b); });
std::ios::sync_with_stdio(false);
std::cin.tie(nullptr);
uint32_t n;
uint64_t m;
std::cin >> n >> m;
std::vector<mint> c(K_MAX + 1);
for (uint32_t i = 0; i < n; ++i) {
uint32_t k;
std::cin >> k;
++c[k];
}
suisen::factorial<mint> fac(n + K_MAX);
mint ans = 0;
suisen::DequeAggregation<mint, op, e> dq;
for (uint32_t d = 0; d < n; ++d) dq.push_front(m + d);
for (uint32_t k = 1; k <= K_MAX; ++k) {
std::vector<mint> e = suisen::eulerian_number<mint>(k);
dq.push_front(m + n + k - 1);
mint sum = 0;
const uint32_t p = std::min(uint64_t(k), m);
for (uint32_t i = 0; i < p; ++i) {
sum += e[i] * dq.prod();
dq.pop_front();
dq.push_back(m - i - 1);
}
ans += c[k] * sum * fac.fac_inv(n + k);
for (uint32_t i = p; i --> 0;) {
dq.push_front(m - i + n + k - 1);
dq.pop_back();
}
}
std::cout << ans.val() << std::endl;
return 0;
}