#line 2 "fps/stirling-matrix.hpp"
#line 2 "modulo/binomial.hpp"
#include <cassert>
#include <type_traits>
#include <vector>
using namespace std;
// コンストラクタの MAX に 「C(n, r) や fac(n) でクエリを投げる最大の n 」
// を入れると倍速くらいになる
// mod を超えて前計算して 0 割りを踏むバグは対策済み
template <typename T>
struct Binomial {
vector<T> f, g, h;
Binomial(int MAX = 0) {
assert(T::get_mod() != 0 && "Binomial<mint>()");
f.resize(1, T{1});
g.resize(1, T{1});
h.resize(1, T{1});
if (MAX > 0) extend(MAX + 1);
}
void extend(int m = -1) {
int n = f.size();
if (m == -1) m = n * 2;
m = min<int>(m, T::get_mod());
if (n >= m) return;
f.resize(m);
g.resize(m);
h.resize(m);
for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i);
g[m - 1] = f[m - 1].inverse();
h[m - 1] = g[m - 1] * f[m - 2];
for (int i = m - 2; i >= n; i--) {
g[i] = g[i + 1] * T(i + 1);
h[i] = g[i] * f[i - 1];
}
}
T fac(int i) {
if (i < 0) return T(0);
while (i >= (int)f.size()) extend();
return f[i];
}
T finv(int i) {
if (i < 0) return T(0);
while (i >= (int)g.size()) extend();
return g[i];
}
T inv(int i) {
if (i < 0) return -inv(-i);
while (i >= (int)h.size()) extend();
return h[i];
}
T C(int n, int r) {
if (n < 0 || n < r || r < 0) return T(0);
return fac(n) * finv(n - r) * finv(r);
}
inline T operator()(int n, int r) { return C(n, r); }
template <typename I>
T multinomial(const vector<I>& r) {
static_assert(is_integral<I>::value == true);
int n = 0;
for (auto& x : r) {
if (x < 0) return T(0);
n += x;
}
T res = fac(n);
for (auto& x : r) res *= finv(x);
return res;
}
template <typename I>
T operator()(const vector<I>& r) {
return multinomial(r);
}
T C_naive(int n, int r) {
if (n < 0 || n < r || r < 0) return T(0);
T ret = T(1);
r = min(r, n - r);
for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--);
return ret;
}
T P(int n, int r) {
if (n < 0 || n < r || r < 0) return T(0);
return fac(n) * finv(n - r);
}
// [x^r] 1 / (1-x)^n
T H(int n, int r) {
if (n < 0 || r < 0) return T(0);
return r == 0 ? 1 : C(n + r - 1, r);
}
};
#line 2 "fps/composite-exp.hpp"
#line 4 "fps/composite-exp.hpp"
#include <utility>
#line 6 "fps/composite-exp.hpp"
using namespace std;
#line 2 "fps/formal-power-series.hpp"
#include <algorithm>
#line 5 "fps/formal-power-series.hpp"
#include <cstdint>
#include <iterator>
#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 9 "fps/composite-exp.hpp"
// 多項式 f に exp(cx) 代入
// 次数 : mod x^{deg} まで計算, 指定がない場合 f と同じ長さ計算
template <typename mint>
FormalPowerSeries<mint> composite_exp(FormalPowerSeries<mint> f, mint c = 1,
int deg = -1) {
using fps = FormalPowerSeries<mint>;
assert(c != 0);
if (deg == -1) deg = f.size();
if (f.empty()) return {};
int N = f.size();
vector<pair<fps, fps>> fs;
for (int i = 0; i < N; i++) fs.emplace_back(fps{f[i]}, fps{1, -c * i});
while (fs.size() > 1u) {
vector<pair<fps, fps>> nx;
for (int i = 0; i + 1 < (int)fs.size(); i += 2) {
pair<fps, fps>& f0 = fs[i];
pair<fps, fps>& f1 = fs[i + 1];
fps s = f0.first * f1.second + f1.first * f0.second;
fps t = f0.second * f1.second;
nx.emplace_back(s, t);
}
if (fs.size() % 2) nx.push_back(fs.back());
fs = nx;
}
fps g = (fs[0].first * fs[0].second.inv(deg)).pre(deg);
mint b = 1;
for (int i = 0; i < deg; i++) g[i] *= b, b /= i + 1;
return g;
}
// 入力 f(x) = sum_{0 <= k < N} a_i exp(ckx) を満たす g(x) (mod x^N)
// 出力 a(x) = sum_{0 <= k < N} a_i x^i
template <typename mint>
FormalPowerSeries<mint> inverse_of_composite_exp(FormalPowerSeries<mint> f,
mint c = 1) {
using fps = FormalPowerSeries<mint>;
if (f.empty()) return {};
int N = f.size();
mint b = 1;
for (int i = 0; i < N; i++) f[i] *= b, b *= i + 1;
int B = 1;
while (B < N) B *= 2;
vector<fps> mod(2 * B, fps{1});
for (int i = 0; i < N; i++) mod[B + i] = fps{-c * i, 1};
for (int i = B - 1; i; i--) mod[i] = mod[2 * i] * mod[2 * i + 1];
fps denom = mod[1].rev();
fps numerator = (f * denom).pre(N);
vector<mint> a(N);
auto dfs = [&](auto rc, int i, int l, int r, fps g) -> void {
if (N <= l) return;
if (l + 1 == r) {
a[l] = g.eval(0);
return;
}
int m = (l + r) / 2;
rc(rc, i * 2 + 0, l, m, g % mod[i * 2 + 0]);
rc(rc, i * 2 + 1, m, r, g % mod[i * 2 + 1]);
};
dfs(dfs, 1, 0, B, numerator.rev());
vector<mint> fac(N);
fac[0] = 1;
for (int i = 1; i < N; i++) fac[i] = fac[i - 1] * c * i;
for (int i = 0; i < N; i++) {
a[i] /= fac[N - 1 - i] * fac[i] * ((N - 1 - i) % 2 ? -1 : 1);
}
return fps{begin(a), end(a)};
}
/**
* @brief $f(exp(cx))$ の計算
*/
#line 2 "fps/multipoint-evaluation.hpp"
#line 4 "fps/multipoint-evaluation.hpp"
template <typename mint>
struct ProductTree {
using fps = FormalPowerSeries<mint>;
const vector<mint> &xs;
vector<fps> buf;
int N, xsz;
vector<int> l, r;
ProductTree(const vector<mint> &xs_) : xs(xs_), xsz(xs.size()) {
N = 1;
while (N < (int)xs.size()) N *= 2;
buf.resize(2 * N);
l.resize(2 * N, xs.size());
r.resize(2 * N, xs.size());
fps::set_fft();
if (fps::ntt_ptr == nullptr)
build();
else
build_ntt();
}
void build() {
for (int i = 0; i < xsz; i++) {
l[i + N] = i;
r[i + N] = i + 1;
buf[i + N] = {-xs[i], 1};
}
for (int i = N - 1; i > 0; i--) {
l[i] = l[(i << 1) | 0];
r[i] = r[(i << 1) | 1];
if (buf[(i << 1) | 0].empty())
continue;
else if (buf[(i << 1) | 1].empty())
buf[i] = buf[(i << 1) | 0];
else
buf[i] = buf[(i << 1) | 0] * buf[(i << 1) | 1];
}
}
void build_ntt() {
fps f;
f.reserve(N * 2);
for (int i = 0; i < xsz; i++) {
l[i + N] = i;
r[i + N] = i + 1;
buf[i + N] = {-xs[i] + 1, -xs[i] - 1};
}
for (int i = N - 1; i > 0; i--) {
l[i] = l[(i << 1) | 0];
r[i] = r[(i << 1) | 1];
if (buf[(i << 1) | 0].empty())
continue;
else if (buf[(i << 1) | 1].empty())
buf[i] = buf[(i << 1) | 0];
else if (buf[(i << 1) | 0].size() == buf[(i << 1) | 1].size()) {
buf[i] = buf[(i << 1) | 0];
f.clear();
copy(begin(buf[(i << 1) | 1]), end(buf[(i << 1) | 1]),
back_inserter(f));
buf[i].ntt_doubling();
f.ntt_doubling();
for (int j = 0; j < (int)buf[i].size(); j++) buf[i][j] *= f[j];
} else {
buf[i] = buf[(i << 1) | 0];
f.clear();
copy(begin(buf[(i << 1) | 1]), end(buf[(i << 1) | 1]),
back_inserter(f));
buf[i].ntt_doubling();
f.intt();
f.resize(buf[i].size(), mint(0));
f.ntt();
for (int j = 0; j < (int)buf[i].size(); j++) buf[i][j] *= f[j];
}
}
for (int i = 0; i < 2 * N; i++) {
buf[i].intt();
buf[i].shrink();
}
}
};
template <typename mint>
vector<mint> InnerMultipointEvaluation(const FormalPowerSeries<mint> &f,
const vector<mint> &xs,
const ProductTree<mint> &ptree) {
using fps = FormalPowerSeries<mint>;
vector<mint> ret;
ret.reserve(xs.size());
auto rec = [&](auto self, fps a, int idx) {
if (ptree.l[idx] == ptree.r[idx]) return;
a %= ptree.buf[idx];
if ((int)a.size() <= 64) {
for (int i = ptree.l[idx]; i < ptree.r[idx]; i++)
ret.push_back(a.eval(xs[i]));
return;
}
self(self, a, (idx << 1) | 0);
self(self, a, (idx << 1) | 1);
};
rec(rec, f, 1);
return ret;
}
template <typename mint>
vector<mint> MultipointEvaluation(const FormalPowerSeries<mint> &f,
const vector<mint> &xs) {
if(f.empty() || xs.empty()) return vector<mint>(xs.size(), mint(0));
return InnerMultipointEvaluation(f, xs, ProductTree<mint>(xs));
}
/**
* @brief Multipoint Evaluation
*/
#line 2 "fps/pascal-matrix.hpp"
#line 5 "fps/pascal-matrix.hpp"
// P_{i, j} = binom(i, j) を満たす行列 P を縦ベクトルに作用
template <typename mint>
FormalPowerSeries<mint> pascal_matrix(FormalPowerSeries<mint> a,
int rev = false) {
using fps = FormalPowerSeries<mint>;
if (a.empty()) return {};
int N = a.size();
Binomial<mint> binom(N + 10);
if (rev == false) {
fps e(N);
for (int i = 0; i < N; i++) {
a[i] *= binom.finv(i);
e[i] = binom.finv(i);
}
fps b = (a * e).pre(N);
for (int i = 0; i < N; i++) b[i] *= binom.fac(i);
return b;
} else {
fps ie(N);
for (int i = 0; i < N; i++) {
a[i] *= binom.finv(i);
ie[i] = binom.finv(i) * (i % 2 ? -1 : 1);
}
fps b = (a * ie).pre(N);
for (int i = 0; i < N; i++) b[i] *= binom.fac(i);
return b;
}
}
// P_{i, j} = binom(j, i) を満たす行列 P を縦ベクトルに作用
template <typename mint>
FormalPowerSeries<mint> pascal_matrix_trans(FormalPowerSeries<mint> a,
int rev = false) {
using fps = FormalPowerSeries<mint>;
if (a.empty()) return {};
int N = a.size();
Binomial<mint> binom(N + 10);
if (rev == false) {
fps e(N);
for (int i = 0; i < N; i++) {
a[i] *= binom.fac(i);
e[i] = binom.finv(i);
}
fps b = (a.rev() * e).pre(N).rev();
for (int i = 0; i < N; i++) b[i] *= binom.finv(i);
return b;
} else {
fps ie(N);
for (int i = 0; i < N; i++) {
a[i] *= binom.fac(i);
ie[i] = binom.finv(i) * (i % 2 ? -1 : 1);
}
fps b = (a.rev() * ie).pre(N).rev();
for (int i = 0; i < N; i++) b[i] *= binom.finv(i);
return b;
}
}
#line 2 "fps/polynomial-interpolation.hpp"
#line 5 "fps/polynomial-interpolation.hpp"
template <class mint>
FormalPowerSeries<mint> PolynomialInterpolation(const vector<mint> &xs,
const vector<mint> &ys) {
using fps = FormalPowerSeries<mint>;
assert(xs.size() == ys.size());
ProductTree<mint> ptree(xs);
fps w = ptree.buf[1].diff();
vector<mint> vs = InnerMultipointEvaluation<mint>(w, xs, ptree);
auto rec = [&](auto self, int idx) -> fps {
if (idx >= ptree.N) {
if (idx - ptree.N < (int)xs.size())
return {ys[idx - ptree.N] / vs[idx - ptree.N]};
else
return {mint(1)};
}
if (ptree.buf[idx << 1 | 0].empty())
return {};
else if (ptree.buf[idx << 1 | 1].empty())
return self(self, idx << 1 | 0);
return self(self, idx << 1 | 0) * ptree.buf[idx << 1 | 1] +
self(self, idx << 1 | 1) * ptree.buf[idx << 1 | 0];
};
return rec(rec, 1);
}
#line 9 "fps/stirling-matrix.hpp"
// S_{i, j} = stirling{i, j} を満たす行列 S を縦ベクトルに作用
template <typename mint>
FormalPowerSeries<mint> stirling_matrix(FormalPowerSeries<mint> a,
int rev = false) {
using fps = FormalPowerSeries<mint>;
if (a.empty()) return {};
int N = a.size();
Binomial<mint> binom(N + 10);
if (rev == false) {
for (int i = 0; i < N; i++) a[i] *= binom.finv(i);
fps f = pascal_matrix_trans(a, true);
fps b = composite_exp<mint>(f, 1, N);
for (int i = 0; i < N; i++) b[i] *= binom.fac(i);
return b;
} else {
for (int i = 0; i < N; i++) a[i] *= binom.finv(i);
fps f = inverse_of_composite_exp<mint>(a, 1);
fps b = pascal_matrix_trans(f, false);
for (int i = 0; i < N; i++) b[i] *= binom.fac(i);
return b;
}
}
// S_{i, j} = stirling{j, i} を満たす行列 S を縦ベクトルに作用
template <typename mint>
FormalPowerSeries<mint> stirling_matrix_trans(FormalPowerSeries<mint> a,
int rev = false) {
using fps = FormalPowerSeries<mint>;
if (a.empty()) return {};
int N = a.size();
Binomial<mint> binom(N + 10);
vector<mint> xs(N);
for (int i = 0; i < N; i++) xs[i] = i;
if (rev == false) {
auto _f = MultipointEvaluation(a, xs);
fps f{begin(_f), end(_f)};
fps g = pascal_matrix(f, true);
for (int i = 0; i < N; i++) g[i] *= binom.finv(i);
return g;
} else {
for (int i = 0; i < N; i++) a[i] *= binom.fac(i);
auto g = pascal_matrix(a, false);
return PolynomialInterpolation(xs, g);
}
}
fps/stirling-matrix.hpp