#pragma once
#include <cassert>
#include <functional>
#include <type_traits>
#include <utility>
#include <vector>
using namespace std;
#include "../internal/internal-function.hpp"
#include "../modint/montgomery-modint.hpp"
#include "ntt-friendly-fps.hpp"
using mint = LazyMontgomeryModInt<998244353>;
using fps = FormalPowerSeries<mint>;
struct ofpsBase {
using ob = ofpsBase;
using Func = nyaan_internal::inplace_function<mint(int), 64>;
Func func;
fps f;
ofpsBase() {
func = [](int) -> mint { return 0; };
}
ofpsBase(const fps& _f) : f(_f) {
func = [this](int i) { return i < (int)f.size() ? f[i] : 0; };
}
ofpsBase(const Func& _func) : func(_func) {}
template <typename F, typename = enable_if_t<is_invocable_r_v<mint, F&, int>>>
ofpsBase(F&& _func) : func(std::forward<F>(_func)) {}
ofpsBase(const ob& rhs) = delete;
ob& operator=(const ob& rhs) = delete;
ofpsBase(ob&& rhs) noexcept = delete;
ob& operator=(ob&& rhs) noexcept = delete;
void set_corner(const fps& _f) { f = _f; }
void set_func(const Func& _func) { func = _func; }
template <typename F>
auto set_func(F&& _func) -> enable_if_t<is_invocable_r_v<mint, F&, int>> {
func = std::forward<F>(_func);
}
mint get(int i) {
while ((int)f.size() <= i) f.push_back(std::invoke(func, f.size()));
return f[i];
}
ob integral() {
return ob{[this](int i) { return i == 0 ? 0 : get(i - 1) / i; }};
}
ob diff() {
return ob{[this](int i) { return get(i + 1) * (i + 1); }};
}
ob operator>>(int s) {
return ob{[this, s](int i) { return get(i + s); }};
}
ob operator<<(int s) {
return ob{[this, s](int i) { return i < s ? 0 : get(i - s); }};
}
friend ob _dot(ob& a, ob& b) {
return ob{[&a, &b](int i) { return a.get(i) * b.get(i); }};
}
friend ob operator+(ob& a, ob& b) {
return ob{[&a, &b](int i) { return a.get(i) + b.get(i); }};
}
friend ob operator-(ob& a, ob& b) {
return ob{[&a, &b](int i) { return a.get(i) - b.get(i); }};
}
friend ob operator*(ob& a, mint b) {
return ob{[&a, b](int i) { return a.get(i) * b; }};
}
friend ob operator*(mint b, ob& a) {
return ob{[&a, b](int i) { return a.get(i) * b; }};
}
};
struct ofpsMul : public ofpsBase {
vector<mint> a, b, c;
fps F, G;
vector<fps> as, bs;
ofpsMul(ob& _a, ob& _b) {
func = [&_a, &_b, this](int idx) -> mint {
while ((int)a.size() <= idx) {
int q = a.size();
a.push_back(_a.get(q)), b.push_back(_b.get(q));
if ((int)c.size() <= q) c.resize(q + 1);
c[q] += a[q] * b[0] + (q ? b[q] * a[0] : 0);
auto precalc = [&](int lg) {
if ((int)as.size() <= lg) as.resize(lg + 1), bs.resize(lg + 1);
if (!as[lg].empty()) return;
int d = 1 << lg;
fps s{begin(a), begin(a) + d * 2};
fps t{begin(b), begin(b) + d * 2};
s.ntt(), t.ntt();
as[lg] = s, bs[lg] = t;
};
q++;
for (int d = 1, lg = 0; d <= q; d *= 2, lg++) {
if (q % (2 * d) != d) continue;
if (q == d) {
F.assign(2 * d, mint{});
G.assign(2 * d, mint{});
for (int i = 0; i < d; i++) F[i] = a[i];
for (int i = 0; i < d; i++) G[i] = b[i];
F.ntt(), G.ntt();
for (int i = 0; i < d * 2; i++) F[i] *= G[i];
F.intt();
if ((int)c.size() < q + d) c.resize(q + d);
for (int i = q; i < q + d; i++) c[i] += F[d + i - q];
} else {
precalc(lg);
F.assign(2 * d, mint{});
G.assign(2 * d, mint{});
for (int i = 0; i < d; i++) F[i] = a[q - d + i];
for (int i = 0; i < d; i++) G[i] = b[q - d + i];
F.ntt(), G.ntt();
fps& s = as[lg];
fps& t = bs[lg];
for (int i = 0; i < d * 2; i++) F[i] = F[i] * t[i] + G[i] * s[i];
F.intt();
if ((int)c.size() < q + d) c.resize(q + d);
for (int i = q; i < q + d; i++) c[i] += F[d + i - q];
}
}
}
return c[idx];
};
}
};
struct ofpsInv : public ofpsBase {
mint oi;
ob& a;
ob b;
ofpsMul s;
ofpsInv(ob& _a)
: a(_a), b([this](int j) { return a.get(j + 1); }), s(b, *this) {
func = [this](int j) -> mint {
assert(a.get(0) != 0);
if (j == 0) return oi = a.get(0).inverse();
get(0);
return j ? s.get(j - 1) * -oi : oi;
};
}
};
struct ofpsExp : public ofpsBase {
ob& a;
ob b;
ofpsMul m;
ofpsExp(ob& _a)
: a(_a),
b([this](int i) { return a.get(i + 1) * (i + 1); }),
m(b, *this) {
func = [this](int i) { return i == 0 ? 1 : m.get(i - 1) / i; };
}
};
struct OnlineFormalPowerSeries {
using ob = ofpsBase;
using ofps = OnlineFormalPowerSeries;
ob* p;
OnlineFormalPowerSeries() : p(new ob()) {}
OnlineFormalPowerSeries(const fps& f) : p(new ob(f)) {}
OnlineFormalPowerSeries(ob* q) : p(q) {}
void set_corner(const fps& f) { p->set_corner(f); }
void set(const ofps& f) { p->set_func(f.p->func); }
mint operator[](int i) { return p->get(i); }
fps pre(int n) {
p->get(n - 1);
return p->f.pre(n);
}
ofps integral() { return new ob{p->integral()}; }
ofps diff() { return new ob{p->diff()}; }
ofps operator>>(int s) { return new ob{(*p) >> s}; }
ofps operator<<(int s) { return new ob{(*p) << s}; }
friend ofps dot(ofps a, ofps b) { return new ob{_dot(*a.p, *b.p)}; }
friend ofps operator+(ofps a, ofps b) { return new ob{*a.p + *b.p}; }
friend ofps operator-(ofps a, ofps b) { return new ob{*a.p - *b.p}; }
friend ofps operator*(ofps a, mint b) { return new ob{*a.p * b}; }
friend ofps operator*(mint a, ofps b) { return new ob{a * *b.p}; }
ofps operator-() { return new ob{*p * -1}; }
friend ofps operator*(ofps a, ofps b) { return new ofpsMul{*a.p, *b.p}; }
friend ofps operator/(ofps a, ofps b) {
ofps invb = b.inv();
return new ofpsMul{*a.p, *invb.p};
}
ofps inv() { return new ofpsInv{*p}; }
ofps exp() { return new ofpsExp{*p}; }
ofps log() { return (this->diff() / *this).integral(); }
};
using ofps = OnlineFormalPowerSeries;
#line 2 "fps/online-fps.hpp"
#include <cassert>
#include <functional>
#include <type_traits>
#include <utility>
#include <vector>
using namespace std;
#line 2 "internal/internal-function.hpp"
#include <cstddef>
#line 5 "internal/internal-function.hpp"
#include <memory>
#line 8 "internal/internal-function.hpp"
namespace nyaan_internal {
template <class>
class function_ref;
template <class R, class... Args>
class function_ref<R(Args...)> {
void* obj_ = nullptr;
R (*call_obj_)(void*, Args...) = nullptr;
R (*func_)(Args...) = nullptr;
public:
function_ref() noexcept = default;
function_ref(std::nullptr_t) noexcept {}
function_ref(R (*f)(Args...)) noexcept : func_(f) {}
template <
class F, class Fn = std::remove_reference_t<F>,
class = std::enable_if_t<
std::is_lvalue_reference_v<F&&> &&
!std::is_same_v<std::decay_t<F>, function_ref> &&
!std::is_pointer_v<std::decay_t<F>> && !std::is_function_v<Fn> &&
std::is_invocable_r_v<R, Fn&, Args...>>>
function_ref(F&& f) noexcept {
obj_ = const_cast<void*>(static_cast<const void*>(std::addressof(f)));
call_obj_ = [](void* p, Args... args) -> R {
return std::invoke(*static_cast<Fn*>(p), std::forward<Args>(args)...);
};
}
R operator()(Args... args) const {
if (call_obj_) {
return call_obj_(obj_, std::forward<Args>(args)...);
}
if (!func_) throw std::bad_function_call();
return func_(std::forward<Args>(args)...);
}
explicit operator bool() const noexcept {
return call_obj_ != nullptr || func_ != nullptr;
}
};
template <class, std::size_t Capacity = 32,
std::size_t Align = alignof(std::max_align_t)>
class inplace_function;
template <class R, class... Args, std::size_t Capacity, std::size_t Align>
class inplace_function<R(Args...), Capacity, Align> {
using storage_t = typename std::aligned_storage<Capacity, Align>::type;
storage_t storage_;
R (*invoke_)(void*, Args&&...) = nullptr;
void (*copy_)(void*, const void*) = nullptr;
void (*move_)(void*, void*) = nullptr;
void (*destroy_)(void*) = nullptr;
template <class F>
static R invoke_impl(void* p, Args&&... args) {
return std::invoke(*static_cast<F*>(p), std::forward<Args>(args)...);
}
template <class F>
static void copy_impl(void* dst, const void* src) {
new (dst) F(*static_cast<const F*>(src));
}
template <class F>
static void move_impl(void* dst, void* src) {
if constexpr (std::is_move_constructible_v<F>) {
new (dst) F(std::move(*static_cast<F*>(src)));
} else {
new (dst) F(*static_cast<F*>(src));
}
}
template <class F>
static void destroy_impl(void* p) {
static_cast<F*>(p)->~F();
}
template <class F>
void emplace(F&& f) {
using Fn = std::decay_t<F>;
static_assert(std::is_invocable_r_v<R, Fn&, Args...>,
"inplace_function target is not invocable with this signature");
static_assert(sizeof(Fn) <= Capacity,
"inplace_function target is too large; increase Capacity");
static_assert(alignof(Fn) <= Align,
"inplace_function target alignment is too strict; increase Align");
static_assert(std::is_copy_constructible_v<Fn>,
"inplace_function target must be copy constructible");
if constexpr (std::is_pointer_v<Fn>) {
if (f == nullptr) return;
}
if constexpr (std::is_move_constructible_v<Fn> ||
std::is_lvalue_reference_v<F>) {
new (&storage_) Fn(std::forward<F>(f));
} else {
new (&storage_) Fn(f);
}
invoke_ = &invoke_impl<Fn>;
copy_ = ©_impl<Fn>;
move_ = &move_impl<Fn>;
destroy_ = &destroy_impl<Fn>;
}
public:
inplace_function() noexcept = default;
inplace_function(std::nullptr_t) noexcept {}
~inplace_function() { reset(); }
inplace_function(const inplace_function& other) {
if (other) {
other.copy_(&storage_, &other.storage_);
invoke_ = other.invoke_;
copy_ = other.copy_;
move_ = other.move_;
destroy_ = other.destroy_;
}
}
inplace_function(inplace_function&& other) {
if (other) {
other.move_(&storage_, &other.storage_);
invoke_ = other.invoke_;
copy_ = other.copy_;
move_ = other.move_;
destroy_ = other.destroy_;
other.reset();
}
}
template <
class F, class Fn = std::decay_t<F>,
class = std::enable_if_t<!std::is_same_v<Fn, inplace_function> &&
!std::is_same_v<Fn, std::nullptr_t>>>
inplace_function(F&& f) {
emplace(std::forward<F>(f));
}
inplace_function& operator=(const inplace_function& other) {
if (this == &other) return *this;
reset();
if (other) {
other.copy_(&storage_, &other.storage_);
invoke_ = other.invoke_;
copy_ = other.copy_;
move_ = other.move_;
destroy_ = other.destroy_;
}
return *this;
}
inplace_function& operator=(inplace_function&& other) {
if (this == &other) return *this;
reset();
if (other) {
other.move_(&storage_, &other.storage_);
invoke_ = other.invoke_;
copy_ = other.copy_;
move_ = other.move_;
destroy_ = other.destroy_;
other.reset();
}
return *this;
}
template <
class F, class Fn = std::decay_t<F>,
class = std::enable_if_t<!std::is_same_v<Fn, inplace_function> &&
!std::is_same_v<Fn, std::nullptr_t>>>
inplace_function& operator=(F&& f) {
reset();
emplace(std::forward<F>(f));
return *this;
}
inplace_function& operator=(std::nullptr_t) noexcept {
reset();
return *this;
}
void reset() noexcept {
if (destroy_) destroy_(&storage_);
invoke_ = nullptr;
copy_ = nullptr;
move_ = nullptr;
destroy_ = nullptr;
}
explicit operator bool() const noexcept { return invoke_ != nullptr; }
R operator()(Args... args) const {
if (!invoke_) throw std::bad_function_call();
return invoke_(
const_cast<void*>(static_cast<const void*>(&storage_)),
std::forward<Args>(args)...);
}
};
} // namespace nyaan_internal
using nyaan_internal::function_ref;
using nyaan_internal::inplace_function;
#line 2 "modint/montgomery-modint.hpp"
#include <cstdint>
#include <iostream>
template <uint32_t mod>
struct LazyMontgomeryModInt {
using mint = LazyMontgomeryModInt;
using i32 = int32_t;
using u32 = uint32_t;
using u64 = uint64_t;
static constexpr u32 get_r() {
u32 ret = mod;
for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret;
return ret;
}
static constexpr u32 r = get_r();
static constexpr u32 n2 = -u64(mod) % mod;
static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
static_assert(r * mod == 1, "this code has bugs.");
u32 a;
constexpr LazyMontgomeryModInt() : a(0) {}
constexpr LazyMontgomeryModInt(const int64_t &b)
: a(reduce(u64(b % mod + mod) * n2)){};
static constexpr u32 reduce(const u64 &b) {
return (b + u64(u32(b) * u32(-r)) * mod) >> 32;
}
constexpr mint &operator+=(const mint &b) {
if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod;
return *this;
}
constexpr mint &operator-=(const mint &b) {
if (i32(a -= b.a) < 0) a += 2 * mod;
return *this;
}
constexpr mint &operator*=(const mint &b) {
a = reduce(u64(a) * b.a);
return *this;
}
constexpr mint &operator/=(const mint &b) {
*this *= b.inverse();
return *this;
}
constexpr mint operator+(const mint &b) const { return mint(*this) += b; }
constexpr mint operator-(const mint &b) const { return mint(*this) -= b; }
constexpr mint operator*(const mint &b) const { return mint(*this) *= b; }
constexpr mint operator/(const mint &b) const { return mint(*this) /= b; }
constexpr bool operator==(const mint &b) const {
return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a);
}
constexpr bool operator!=(const mint &b) const {
return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a);
}
constexpr mint operator-() const { return mint() - mint(*this); }
constexpr mint operator+() const { return mint(*this); }
constexpr mint pow(u64 n) const {
mint ret(1), mul(*this);
while (n > 0) {
if (n & 1) ret *= mul;
mul *= mul;
n >>= 1;
}
return ret;
}
constexpr mint inverse() const {
int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0;
while (y > 0) {
t = x / y;
x -= t * y, u -= t * v;
tmp = x, x = y, y = tmp;
tmp = u, u = v, v = tmp;
}
return mint{u};
}
friend std::ostream &operator<<(std::ostream &os, const mint &b) {
return os << b.get();
}
friend std::istream &operator>>(std::istream &is, mint &b) {
int64_t t;
is >> t;
b = LazyMontgomeryModInt<mod>(t);
return (is);
}
constexpr u32 get() const {
u32 ret = reduce(a);
return ret >= mod ? ret - mod : ret;
}
static constexpr u32 get_mod() { return mod; }
};
#line 2 "fps/ntt-friendly-fps.hpp"
#line 2 "ntt/ntt.hpp"
#include <algorithm>
#line 5 "ntt/ntt.hpp"
#include <iterator>
#line 7 "ntt/ntt.hpp"
using namespace std;
template <typename mint>
struct NTT {
static constexpr uint32_t get_pr() {
uint32_t _mod = mint::get_mod();
using u64 = uint64_t;
u64 ds[32] = {};
int idx = 0;
u64 m = _mod - 1;
for (u64 i = 2; i * i <= m; ++i) {
if (m % i == 0) {
ds[idx++] = i;
while (m % i == 0) m /= i;
}
}
if (m != 1) ds[idx++] = m;
uint32_t _pr = 2;
while (1) {
int flg = 1;
for (int i = 0; i < idx; ++i) {
u64 a = _pr, b = (_mod - 1) / ds[i], r = 1;
while (b) {
if (b & 1) r = r * a % _mod;
a = a * a % _mod;
b >>= 1;
}
if (r == 1) {
flg = 0;
break;
}
}
if (flg == 1) break;
++_pr;
}
return _pr;
};
static constexpr uint32_t mod = mint::get_mod();
static constexpr uint32_t pr = get_pr();
static constexpr int level = __builtin_ctzll(mod - 1);
mint dw[level], dy[level];
void setwy(int k) {
mint w[level], y[level];
w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
y[k - 1] = w[k - 1].inverse();
for (int i = k - 2; i > 0; --i)
w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1];
dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2];
for (int i = 3; i < k; ++i) {
dw[i] = dw[i - 1] * y[i - 2] * w[i];
dy[i] = dy[i - 1] * w[i - 2] * y[i];
}
}
NTT() { setwy(level); }
void fft4(vector<mint> &a, int k) {
if ((int)a.size() <= 1) return;
if (k == 1) {
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
if (k & 1) {
int v = 1 << (k - 1);
for (int j = 0; j < v; ++j) {
mint ajv = a[j + v];
a[j + v] = a[j] - ajv;
a[j] += ajv;
}
}
int u = 1 << (2 + (k & 1));
int v = 1 << (k - 2 - (k & 1));
mint one = mint(1);
mint imag = dw[1];
while (v) {
// jh = 0
{
int j0 = 0;
int j1 = v;
int j2 = j1 + v;
int j3 = j2 + v;
for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
mint t0p2 = t0 + t2, t1p3 = t1 + t3;
mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3;
}
}
// jh >= 1
mint ww = one, xx = one * dw[2], wx = one;
for (int jh = 4; jh < u;) {
ww = xx * xx, wx = ww * xx;
int j0 = jh * v;
int je = j0 + v;
int j2 = je + v;
for (; j0 < je; ++j0, ++j2) {
mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww,
t3 = a[j2 + v] * wx;
mint t0p2 = t0 + t2, t1p3 = t1 + t3;
mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag;
a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3;
a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3;
}
xx *= dw[__builtin_ctzll((jh += 4))];
}
u <<= 2;
v >>= 2;
}
}
void ifft4(vector<mint> &a, int k) {
if ((int)a.size() <= 1) return;
if (k == 1) {
mint a1 = a[1];
a[1] = a[0] - a[1];
a[0] = a[0] + a1;
return;
}
int u = 1 << (k - 2);
int v = 1;
mint one = mint(1);
mint imag = dy[1];
while (u) {
// jh = 0
{
int j0 = 0;
int j1 = v;
int j2 = v + v;
int j3 = j2 + v;
for (; j0 < v; ++j0, ++j1, ++j2, ++j3) {
mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
mint t0p1 = t0 + t1, t2p3 = t2 + t3;
mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag;
a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3;
a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3;
}
}
// jh >= 1
mint ww = one, xx = one * dy[2], yy = one;
u <<= 2;
for (int jh = 4; jh < u;) {
ww = xx * xx, yy = xx * imag;
int j0 = jh * v;
int je = j0 + v;
int j2 = je + v;
for (; j0 < je; ++j0, ++j2) {
mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
mint t0p1 = t0 + t1, t2p3 = t2 + t3;
mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy;
a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww;
a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww;
}
xx *= dy[__builtin_ctzll(jh += 4)];
}
u >>= 4;
v <<= 2;
}
if (k & 1) {
u = 1 << (k - 1);
for (int j = 0; j < u; ++j) {
mint ajv = a[j] - a[j + u];
a[j] += a[j + u];
a[j + u] = ajv;
}
}
}
void ntt(vector<mint> &a) {
if ((int)a.size() <= 1) return;
fft4(a, __builtin_ctz(a.size()));
}
void intt(vector<mint> &a) {
if ((int)a.size() <= 1) return;
ifft4(a, __builtin_ctz(a.size()));
mint iv = mint(a.size()).inverse();
for (auto &x : a) x *= iv;
}
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b) {
int l = a.size() + b.size() - 1;
if (min<int>(a.size(), b.size()) <= 40) {
vector<mint> s(l);
for (int i = 0; i < (int)a.size(); ++i)
for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j];
return s;
}
int k = 2, M = 4;
while (M < l) M <<= 1, ++k;
setwy(k);
vector<mint> s(M);
for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i];
fft4(s, k);
if (a.size() == b.size() && a == b) {
for (int i = 0; i < M; ++i) s[i] *= s[i];
} else {
vector<mint> t(M);
for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i];
fft4(t, k);
for (int i = 0; i < M; ++i) s[i] *= t[i];
}
ifft4(s, k);
s.resize(l);
mint invm = mint(M).inverse();
for (int i = 0; i < l; ++i) s[i] *= invm;
return s;
}
void ntt_doubling(vector<mint> &a) {
int M = (int)a.size();
auto b = a;
intt(b);
mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1));
for (int i = 0; i < M; i++) b[i] *= r, r *= zeta;
ntt(b);
copy(begin(b), end(b), back_inserter(a));
}
};
#line 2 "fps/formal-power-series.hpp"
#line 8 "fps/formal-power-series.hpp"
using namespace std;
template <typename mint>
struct FormalPowerSeries : vector<mint> {
using vector<mint>::vector;
using FPS = FormalPowerSeries;
FPS &operator+=(const FPS &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i];
return *this;
}
FPS &operator+=(const mint &r) {
if (this->empty()) this->resize(1);
(*this)[0] += r;
return *this;
}
FPS &operator-=(const FPS &r) {
if (r.size() > this->size()) this->resize(r.size());
for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i];
return *this;
}
FPS &operator-=(const mint &r) {
if (this->empty()) this->resize(1);
(*this)[0] -= r;
return *this;
}
FPS &operator*=(const mint &v) {
for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v;
return *this;
}
FPS &operator/=(const FPS &r) {
if (this->size() < r.size()) {
this->clear();
return *this;
}
int n = this->size() - r.size() + 1;
if ((int)r.size() <= 64) {
FPS f(*this), g(r);
g.shrink();
mint coeff = g.back().inverse();
for (auto &x : g) x *= coeff;
int deg = (int)f.size() - (int)g.size() + 1;
int gs = g.size();
FPS quo(deg);
for (int i = deg - 1; i >= 0; i--) {
quo[i] = f[i + gs - 1];
for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j];
}
*this = quo * coeff;
this->resize(n, mint(0));
return *this;
}
return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev();
}
FPS &operator%=(const FPS &r) {
*this -= *this / r * r;
shrink();
return *this;
}
FPS operator+(const FPS &r) const { return FPS(*this) += r; }
FPS operator+(const mint &v) const { return FPS(*this) += v; }
FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
FPS operator-(const mint &v) const { return FPS(*this) -= v; }
FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
FPS operator*(const mint &v) const { return FPS(*this) *= v; }
FPS operator/(const FPS &r) const { return FPS(*this) /= r; }
FPS operator%(const FPS &r) const { return FPS(*this) %= r; }
FPS operator-() const {
FPS ret(this->size());
for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i];
return ret;
}
void shrink() {
while (this->size() && this->back() == mint(0)) this->pop_back();
}
FPS rev() const {
FPS ret(*this);
reverse(begin(ret), end(ret));
return ret;
}
FPS dot(FPS r) const {
FPS ret(min(this->size(), r.size()));
for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i];
return ret;
}
// 前 sz 項を取ってくる。sz に足りない項は 0 埋めする
FPS pre(int sz) const {
FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz));
if ((int)ret.size() < sz) ret.resize(sz);
return ret;
}
FPS operator>>(int sz) const {
if ((int)this->size() <= sz) return {};
FPS ret(*this);
ret.erase(ret.begin(), ret.begin() + sz);
return ret;
}
FPS operator<<(int sz) const {
FPS ret(*this);
ret.insert(ret.begin(), sz, mint(0));
return ret;
}
FPS diff() const {
const int n = (int)this->size();
FPS ret(max(0, n - 1));
mint one(1), coeff(1);
for (int i = 1; i < n; i++) {
ret[i - 1] = (*this)[i] * coeff;
coeff += one;
}
return ret;
}
FPS integral() const {
const int n = (int)this->size();
FPS ret(n + 1);
ret[0] = mint(0);
if (n > 0) ret[1] = mint(1);
auto mod = mint::get_mod();
for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i);
for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i];
return ret;
}
mint eval(mint x) const {
mint r = 0, w = 1;
for (auto &v : *this) r += w * v, w *= x;
return r;
}
FPS log(int deg = -1) const {
assert(!(*this).empty() && (*this)[0] == mint(1));
if (deg == -1) deg = (int)this->size();
return (this->diff() * this->inv(deg)).pre(deg - 1).integral();
}
FPS pow(int64_t k, int deg = -1) const {
const int n = (int)this->size();
if (deg == -1) deg = n;
if (k == 0) {
FPS ret(deg);
if (deg) ret[0] = 1;
return ret;
}
for (int i = 0; i < n; i++) {
if ((*this)[i] != mint(0)) {
mint rev = mint(1) / (*this)[i];
FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg);
ret *= (*this)[i].pow(k);
ret = (ret << (i * k)).pre(deg);
if ((int)ret.size() < deg) ret.resize(deg, mint(0));
return ret;
}
if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0));
}
return FPS(deg, mint(0));
}
static void *ntt_ptr;
static void set_fft();
FPS &operator*=(const FPS &r);
void ntt();
void intt();
void ntt_doubling();
static int ntt_pr();
FPS inv(int deg = -1) const;
FPS exp(int deg = -1) const;
};
template <typename mint>
void *FormalPowerSeries<mint>::ntt_ptr = nullptr;
template <int N>
struct FPSBackendPriority : FPSBackendPriority<N - 1> {};
template <>
struct FPSBackendPriority<0> {};
template <typename mint>
void FormalPowerSeries<mint>::set_fft() {
fps_set_fft_impl((FormalPowerSeries<mint>*)nullptr, FPSBackendPriority<1>{});
}
template <typename mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(const FPS& r) {
if (this->empty() || r.empty()) {
this->clear();
return *this;
}
return fps_multiply_impl(*this, r, FPSBackendPriority<1>{});
}
template <typename mint>
void FormalPowerSeries<mint>::ntt() {
fps_ntt_impl(*this, FPSBackendPriority<1>{});
}
template <typename mint>
void FormalPowerSeries<mint>::intt() {
fps_intt_impl(*this, FPSBackendPriority<1>{});
}
template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
fps_ntt_doubling_impl(*this, FPSBackendPriority<1>{});
}
template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
return fps_ntt_pr_impl((FormalPowerSeries<mint>*)nullptr,
FPSBackendPriority<1>{});
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
return fps_inv_impl(*this, deg, FPSBackendPriority<1>{});
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::exp(int deg) const {
return fps_exp_impl(*this, deg, FPSBackendPriority<1>{});
}
/**
* @brief 多項式/形式的冪級数ライブラリ
*/
#line 5 "fps/ntt-friendly-fps.hpp"
template <typename mint>
void fps_set_fft_impl(FormalPowerSeries<mint>*, FPSBackendPriority<1>) {
if (!FormalPowerSeries<mint>::ntt_ptr) {
FormalPowerSeries<mint>::ntt_ptr = new NTT<mint>;
}
}
template <typename mint>
FormalPowerSeries<mint>& fps_multiply_impl(FormalPowerSeries<mint>& f,
const FormalPowerSeries<mint>& r,
FPSBackendPriority<1>) {
FormalPowerSeries<mint>::set_fft();
auto ret = static_cast<NTT<mint>*>(FormalPowerSeries<mint>::ntt_ptr)->multiply(f, r);
return f = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <typename mint>
void fps_ntt_impl(FormalPowerSeries<mint>& f, FPSBackendPriority<1>) {
FormalPowerSeries<mint>::set_fft();
static_cast<NTT<mint>*>(FormalPowerSeries<mint>::ntt_ptr)->ntt(f);
}
template <typename mint>
void fps_intt_impl(FormalPowerSeries<mint>& f, FPSBackendPriority<1>) {
FormalPowerSeries<mint>::set_fft();
static_cast<NTT<mint>*>(FormalPowerSeries<mint>::ntt_ptr)->intt(f);
}
template <typename mint>
void fps_ntt_doubling_impl(FormalPowerSeries<mint>& f, FPSBackendPriority<1>) {
FormalPowerSeries<mint>::set_fft();
static_cast<NTT<mint>*>(FormalPowerSeries<mint>::ntt_ptr)->ntt_doubling(f);
}
template <typename mint>
int fps_ntt_pr_impl(FormalPowerSeries<mint>*, FPSBackendPriority<1>) {
FormalPowerSeries<mint>::set_fft();
return static_cast<NTT<mint>*>(FormalPowerSeries<mint>::ntt_ptr)->pr;
}
template <typename mint>
FormalPowerSeries<mint> fps_inv_impl(const FormalPowerSeries<mint>& f, int deg,
FPSBackendPriority<1>) {
assert(f[0] != mint(0));
if (deg == -1) deg = (int)f.size();
FormalPowerSeries<mint> res(deg);
res[0] = {mint(1) / f[0]};
for (int d = 1; d < deg; d <<= 1) {
FormalPowerSeries<mint> F(2 * d), g(2 * d);
for (int j = 0; j < min((int)f.size(), 2 * d); j++) F[j] = f[j];
for (int j = 0; j < d; j++) g[j] = res[j];
F.ntt();
g.ntt();
for (int j = 0; j < 2 * d; j++) F[j] *= g[j];
F.intt();
for (int j = 0; j < d; j++) F[j] = 0;
F.ntt();
for (int j = 0; j < 2 * d; j++) F[j] *= g[j];
F.intt();
for (int j = d; j < min(2 * d, deg); j++) res[j] = -F[j];
}
return res.pre(deg);
}
template <typename mint>
FormalPowerSeries<mint> fps_exp_impl(const FormalPowerSeries<mint>& f, int deg,
FPSBackendPriority<1>) {
using fps = FormalPowerSeries<mint>;
assert(f.size() == 0 || f[0] == mint(0));
if (deg == -1) deg = f.size();
fps inv;
inv.reserve(deg + 1);
inv.push_back(mint(0));
inv.push_back(mint(1));
auto inplace_integral = [&](fps& F) -> void {
const int n = (int)F.size();
auto mod = mint::get_mod();
while ((int)inv.size() <= n) {
int i = inv.size();
inv.push_back((-inv[mod % i]) * (mod / i));
}
F.insert(begin(F), mint(0));
for (int i = 1; i <= n; i++) F[i] *= inv[i];
};
auto inplace_diff = [](fps& F) -> void {
if (F.empty()) return;
F.erase(begin(F));
mint coeff = 1, one = 1;
for (int i = 0; i < (int)F.size(); i++) {
F[i] *= coeff;
coeff += one;
}
};
fps b{1, 1 < (int)f.size() ? f[1] : 0}, c{1}, z1, z2{1, 1};
for (int m = 2; m < deg; m *= 2) {
auto y = b;
y.resize(2 * m);
y.ntt();
z1 = z2;
fps z(m);
for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i];
z.intt();
fill(begin(z), begin(z) + m / 2, mint(0));
z.ntt();
for (int i = 0; i < m; ++i) z[i] *= -z1[i];
z.intt();
c.insert(end(c), begin(z) + m / 2, end(z));
z2 = c;
z2.resize(2 * m);
z2.ntt();
fps x(begin(f), begin(f) + min<int>(f.size(), m));
x.resize(m);
inplace_diff(x);
x.push_back(mint(0));
x.ntt();
for (int i = 0; i < m; ++i) x[i] *= y[i];
x.intt();
x -= b.diff();
x.resize(2 * m);
for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0);
x.ntt();
for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i];
x.intt();
x.pop_back();
inplace_integral(x);
for (int i = m; i < min<int>(f.size(), 2 * m); ++i) x[i] += f[i];
fill(begin(x), begin(x) + m, mint(0));
x.ntt();
for (int i = 0; i < 2 * m; ++i) x[i] *= y[i];
x.intt();
b.insert(end(b), begin(x) + m, end(x));
}
return fps{begin(b), begin(b) + deg};
}
/**
* @brief NTT mod用FPSライブラリ
*/
#line 13 "fps/online-fps.hpp"
using mint = LazyMontgomeryModInt<998244353>;
using fps = FormalPowerSeries<mint>;
struct ofpsBase {
using ob = ofpsBase;
using Func = nyaan_internal::inplace_function<mint(int), 64>;
Func func;
fps f;
ofpsBase() {
func = [](int) -> mint { return 0; };
}
ofpsBase(const fps& _f) : f(_f) {
func = [this](int i) { return i < (int)f.size() ? f[i] : 0; };
}
ofpsBase(const Func& _func) : func(_func) {}
template <typename F, typename = enable_if_t<is_invocable_r_v<mint, F&, int>>>
ofpsBase(F&& _func) : func(std::forward<F>(_func)) {}
ofpsBase(const ob& rhs) = delete;
ob& operator=(const ob& rhs) = delete;
ofpsBase(ob&& rhs) noexcept = delete;
ob& operator=(ob&& rhs) noexcept = delete;
void set_corner(const fps& _f) { f = _f; }
void set_func(const Func& _func) { func = _func; }
template <typename F>
auto set_func(F&& _func) -> enable_if_t<is_invocable_r_v<mint, F&, int>> {
func = std::forward<F>(_func);
}
mint get(int i) {
while ((int)f.size() <= i) f.push_back(std::invoke(func, f.size()));
return f[i];
}
ob integral() {
return ob{[this](int i) { return i == 0 ? 0 : get(i - 1) / i; }};
}
ob diff() {
return ob{[this](int i) { return get(i + 1) * (i + 1); }};
}
ob operator>>(int s) {
return ob{[this, s](int i) { return get(i + s); }};
}
ob operator<<(int s) {
return ob{[this, s](int i) { return i < s ? 0 : get(i - s); }};
}
friend ob _dot(ob& a, ob& b) {
return ob{[&a, &b](int i) { return a.get(i) * b.get(i); }};
}
friend ob operator+(ob& a, ob& b) {
return ob{[&a, &b](int i) { return a.get(i) + b.get(i); }};
}
friend ob operator-(ob& a, ob& b) {
return ob{[&a, &b](int i) { return a.get(i) - b.get(i); }};
}
friend ob operator*(ob& a, mint b) {
return ob{[&a, b](int i) { return a.get(i) * b; }};
}
friend ob operator*(mint b, ob& a) {
return ob{[&a, b](int i) { return a.get(i) * b; }};
}
};
struct ofpsMul : public ofpsBase {
vector<mint> a, b, c;
fps F, G;
vector<fps> as, bs;
ofpsMul(ob& _a, ob& _b) {
func = [&_a, &_b, this](int idx) -> mint {
while ((int)a.size() <= idx) {
int q = a.size();
a.push_back(_a.get(q)), b.push_back(_b.get(q));
if ((int)c.size() <= q) c.resize(q + 1);
c[q] += a[q] * b[0] + (q ? b[q] * a[0] : 0);
auto precalc = [&](int lg) {
if ((int)as.size() <= lg) as.resize(lg + 1), bs.resize(lg + 1);
if (!as[lg].empty()) return;
int d = 1 << lg;
fps s{begin(a), begin(a) + d * 2};
fps t{begin(b), begin(b) + d * 2};
s.ntt(), t.ntt();
as[lg] = s, bs[lg] = t;
};
q++;
for (int d = 1, lg = 0; d <= q; d *= 2, lg++) {
if (q % (2 * d) != d) continue;
if (q == d) {
F.assign(2 * d, mint{});
G.assign(2 * d, mint{});
for (int i = 0; i < d; i++) F[i] = a[i];
for (int i = 0; i < d; i++) G[i] = b[i];
F.ntt(), G.ntt();
for (int i = 0; i < d * 2; i++) F[i] *= G[i];
F.intt();
if ((int)c.size() < q + d) c.resize(q + d);
for (int i = q; i < q + d; i++) c[i] += F[d + i - q];
} else {
precalc(lg);
F.assign(2 * d, mint{});
G.assign(2 * d, mint{});
for (int i = 0; i < d; i++) F[i] = a[q - d + i];
for (int i = 0; i < d; i++) G[i] = b[q - d + i];
F.ntt(), G.ntt();
fps& s = as[lg];
fps& t = bs[lg];
for (int i = 0; i < d * 2; i++) F[i] = F[i] * t[i] + G[i] * s[i];
F.intt();
if ((int)c.size() < q + d) c.resize(q + d);
for (int i = q; i < q + d; i++) c[i] += F[d + i - q];
}
}
}
return c[idx];
};
}
};
struct ofpsInv : public ofpsBase {
mint oi;
ob& a;
ob b;
ofpsMul s;
ofpsInv(ob& _a)
: a(_a), b([this](int j) { return a.get(j + 1); }), s(b, *this) {
func = [this](int j) -> mint {
assert(a.get(0) != 0);
if (j == 0) return oi = a.get(0).inverse();
get(0);
return j ? s.get(j - 1) * -oi : oi;
};
}
};
struct ofpsExp : public ofpsBase {
ob& a;
ob b;
ofpsMul m;
ofpsExp(ob& _a)
: a(_a),
b([this](int i) { return a.get(i + 1) * (i + 1); }),
m(b, *this) {
func = [this](int i) { return i == 0 ? 1 : m.get(i - 1) / i; };
}
};
struct OnlineFormalPowerSeries {
using ob = ofpsBase;
using ofps = OnlineFormalPowerSeries;
ob* p;
OnlineFormalPowerSeries() : p(new ob()) {}
OnlineFormalPowerSeries(const fps& f) : p(new ob(f)) {}
OnlineFormalPowerSeries(ob* q) : p(q) {}
void set_corner(const fps& f) { p->set_corner(f); }
void set(const ofps& f) { p->set_func(f.p->func); }
mint operator[](int i) { return p->get(i); }
fps pre(int n) {
p->get(n - 1);
return p->f.pre(n);
}
ofps integral() { return new ob{p->integral()}; }
ofps diff() { return new ob{p->diff()}; }
ofps operator>>(int s) { return new ob{(*p) >> s}; }
ofps operator<<(int s) { return new ob{(*p) << s}; }
friend ofps dot(ofps a, ofps b) { return new ob{_dot(*a.p, *b.p)}; }
friend ofps operator+(ofps a, ofps b) { return new ob{*a.p + *b.p}; }
friend ofps operator-(ofps a, ofps b) { return new ob{*a.p - *b.p}; }
friend ofps operator*(ofps a, mint b) { return new ob{*a.p * b}; }
friend ofps operator*(mint a, ofps b) { return new ob{a * *b.p}; }
ofps operator-() { return new ob{*p * -1}; }
friend ofps operator*(ofps a, ofps b) { return new ofpsMul{*a.p, *b.p}; }
friend ofps operator/(ofps a, ofps b) {
ofps invb = b.inv();
return new ofpsMul{*a.p, *invb.p};
}
ofps inv() { return new ofpsInv{*p}; }
ofps exp() { return new ofpsExp{*p}; }
ofps log() { return (this->diff() / *this).integral(); }
};
using ofps = OnlineFormalPowerSeries;
fps/online-fps.hpp