#line 1 "verify/set/UNIT_composite_set_power_series.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 2 "template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
#line 2 "template/macro.hpp"
#define rep(i, a, b) for (int i = (a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b) - 1; i >= (a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
#define YN(b) cout << ((b) ? "YES" : "NO") << "\n";
#define Yn(b) cout << ((b) ? "Yes" : "No") << "\n";
#define yn(b) cout << ((b) ? "yes" : "no") << "\n";
#line 6 "template/template.hpp"
#line 2 "template/util.hpp"
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <class T, class S = T>
S SUM(const vector<T> &a) {
return accumulate(ALL(a), S(0));
}
template <class T>
inline bool chmin(T &a, T b) {
if (a > b) {
a = b;
return true;
}
return false;
}
template <class T>
inline bool chmax(T &a, T b) {
if (a < b) {
a = b;
return true;
}
return false;
}
template <class T>
int popcnt(T x) {
return __builtin_popcountll(x);
}
template <class T>
int topbit(T x) {
return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <class T>
int lowbit(T x) {
return (x == 0 ? -1 : __builtin_ctzll(x));
}
#line 8 "template/template.hpp"
#line 2 "template/inout.hpp"
struct Fast {
Fast() {
cin.tie(nullptr);
ios_base::sync_with_stdio(false);
cout << fixed << setprecision(15);
}
} fast;
template <class T1, class T2>
istream &operator>>(istream &is, pair<T1, T2> &p) {
return is >> p.first >> p.second;
}
template <class T1, class T2>
ostream &operator<<(ostream &os, const pair<T1, T2> &p) {
return os << p.first << " " << p.second;
}
template <class T>
istream &operator>>(istream &is, vector<T> &a) {
for (auto &v : a) is >> v;
return is;
}
template <class T>
ostream &operator<<(ostream &os, const vector<T> &a) {
for (auto it = a.begin(); it != a.end();) {
os << *it;
if (++it != a.end()) os << " ";
}
return os;
}
template <class T>
ostream &operator<<(ostream &os, const set<T> &st) {
os << "{";
for (auto it = st.begin(); it != st.end();) {
os << *it;
if (++it != st.end()) os << ",";
}
os << "}";
return os;
}
template <class T1, class T2>
ostream &operator<<(ostream &os, const map<T1, T2> &mp) {
os << "{";
for (auto it = mp.begin(); it != mp.end();) {
os << it->first << ":" << it->second;
if (++it != mp.end()) os << ",";
}
os << "}";
return os;
}
void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
cin >> t;
in(u...);
}
void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
cout << t;
if (sizeof...(u)) cout << sep;
out(u...);
}
#line 10 "template/template.hpp"
#line 2 "template/debug.hpp"
#ifdef LOCAL
#define debug 1
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define debug 0
#define show(...) true
#endif
template <class T>
void _show(int i, T name) {
cerr << '\n';
}
template <class T1, class T2, class... T3>
void _show(int i, const T1 &a, const T2 &b, const T3 &...c) {
for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i];
cerr << ":" << b << " ";
_show(i + 1, a, c...);
}
#line 2 "modint/modint.hpp"
template <unsigned int m = 998244353>
struct ModInt {
using mint = ModInt;
unsigned int _v;
static constexpr unsigned int get_mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
ModInt() : _v(0) {}
ModInt(int64_t v) {
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
unsigned int val() const { return _v; }
mint &operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint &operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint &operator+=(const mint &rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint &operator-=(const mint &rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint &operator*=(const mint &rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
return *this;
}
mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
assert(_v);
return pow(umod() - 2);
}
friend mint operator+(const mint &lhs, const mint &rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint &lhs, const mint &rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint &lhs, const mint &rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint &lhs, const mint &rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint &lhs, const mint &rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint &lhs, const mint &rhs) {
return lhs._v != rhs._v;
}
friend istream &operator>>(istream &is, mint &x) {
return is >> x._v;
}
friend ostream &operator<<(ostream &os, const mint &x) {
return os << x.val();
}
private:
static constexpr unsigned int umod() { return m; }
};
#line 5 "verify/set/UNIT_composite_set_power_series.test.cpp"
using mint = ModInt<998244353>;
#line 2 "set/subset-convolution.hpp"
template <class mint, int n_>
struct SubsetConvolution {
static constexpr int n = n_;
using poly = array<mint, n_ + 1>;
vector<int> pc;
SubsetConvolution() {
pc.assign(1 << n, 0);
for (int i = 1; i < pc.size(); i++) pc[i] = pc[i >> 1] + (i & 1);
}
void poly_add(poly& p, const poly& q, int d) {
for (int i = 0; i < d; i++) p[i] += q[i];
}
void poly_sub(poly& p, const poly& q, int d) {
for (int i = d; i <= n; i++) p[i] -= q[i];
}
void poly_mul(poly& p, const poly& q) {
poly r{};
for (int i = 0; i <= n; i++)
for (int j = 0; j <= n - i; j++)
r[i + j] += p[i] * q[j];
swap(p, r);
}
vector<poly> lift(const vector<mint>& a) {
int n = a.size();
assert(n == (n & -n));
vector<poly> b(n);
for (int i = 0; i < n; i++) {
b[i].fill(0);
b[i][pc[i]] = a[i];
}
return b;
}
vector<mint> unlift(const vector<poly>& b) {
int n = b.size();
assert(n == (n & -n));
vector<mint> a(n);
for (int i = 0; i < n; i++) a[i] = b[i][pc[i]];
return a;
}
void ranked_zeta(vector<poly>& a) {
int n = a.size();
for (int i = 1; i < n; i <<= 1)
for (int j = 0; j < n; j += i * 2)
for (int k = 0; k < i; k++)
poly_add(a[i + j + k], a[j + k], pc[i + j + k]);
}
void ranked_mobius(vector<poly>& a) {
int n = a.size();
for (int i = 1; i < n; i <<= 1)
for (int j = 0; j < n; j += i * 2)
for (int k = 0; k < i; k++)
poly_sub(a[i + j + k], a[j + k], pc[i + j + k]);
}
void ranked_mul(vector<poly>& a, const vector<poly>& b) {
for (int i = 0; i < a.size(); i++) poly_mul(a[i], b[i]);
}
vector<mint> multiply(const vector<mint>& a, const vector<mint>& b) {
auto p = lift(a);
auto q = lift(b);
ranked_zeta(p);
ranked_zeta(q);
ranked_mul(p, q);
ranked_mobius(p);
return unlift(p);
}
};
/**
* @brief Subset Convolution
* @docs docs/set/subset-convolution.md
*/
#line 3 "set/exp-of-set-power-series.hpp"
namespace SetPowerSeries {
template <class mint, int sz = 21>
vector<mint> exp(const vector<mint>& a) {
static SubsetConvolution<mint, sz> sc;
assert(a[0] == 0);
int l = __builtin_ctz(a.size());
assert(a.size() == (1 << l));
vector<mint> f(1 << l, 0);
f[0] = 1;
for (int k = 0; k < l; k++) {
vector<mint> g(f.begin(), f.begin() + (1 << k));
vector<mint> h(a.begin() + (1 << k), a.begin() + (2 << k));
g = sc.multiply(g, h);
copy(g.begin(), g.end(), f.begin() + (1 << k));
}
return f;
}
}; // namespace SetPowerSeries
/**
* @brief Exp Of Set Power Series
*/
#line 3 "set/composite-set-power-series.hpp"
namespace SetPowerSeries {
template <class mint, int sz = 21>
vector<mint> composite_egf(vector<mint> f, vector<mint> a) {
static SubsetConvolution<mint, sz> sc;
assert(a[0] == 0);
if (f.empty()) return vector<mint>(a.size());
int l = __builtin_ctz(a.size());
f.resize(l + 1);
vector<vector<mint>> g(l + 1);
for (int i = 0; i <= l; i++) g[i] = vector<mint>(1 << (l - i), 0);
for (int i = 0; i <= l; i++) g[i][0] = f[i];
for (int k = 0; k < l; k++) {
auto p = sc.lift(vector<mint>(a.begin() + (1 << k), a.begin() + (2 << k)));
sc.ranked_zeta(p);
for (int i = 0; i < l - k; i++) {
auto q = sc.lift(vector<mint>(g[i + 1].begin(), g[i + 1].begin() + (1 << k)));
sc.ranked_zeta(q);
sc.ranked_mul(q, p);
sc.ranked_mobius(q);
auto h = sc.unlift(q);
copy(h.begin(), h.end(), g[i].begin() + (1 << k));
}
}
return g[0];
}
template <class mint, int sz = 21>
vector<mint> composite_polynomial(vector<mint> p, vector<mint> a) {
if (p.empty()) return vector<mint>(a.size());
int l = __builtin_ctz(a.size());
if (a[0] != 0) {
mint c = a[0];
a[0] = 0;
vector<mint> p1(l + 1, 0), binom(l + 1, 0);
binom[0] = 1;
for (int i = 0; i < p.size(); i++) {
mint r = i <= l ? 1 : c.pow(i - l);
for (int j = min(i, l); j >= 0; j--, r *= c) p1[j] += p[i] * binom[j] * r;
for (int j = l; j > 0; j--) binom[j] += binom[j - 1];
}
swap(p, p1);
}
mint r = 1;
for (int i = 1; i <= l; i++) p[i] *= (r *= i);
return composite_egf<mint, sz>(p, a);
}
// log(a), [x^0]a=1
// require inverse of 1,...,sz
template <class mint, int sz = 21>
vector<mint> log(vector<mint> a) {
static SubsetConvolution<mint, sz> sc;
assert(a[0] == 1);
int l = __builtin_ctz(a.size());
if (l == 0) return {0};
vector<mint> inv(l + 1, 1);
rep(i, 1, l + 1) inv[i] = mint(i).inv();
auto p = sc.lift(a);
sc.ranked_zeta(p);
for (int k = 0; k < p.size(); k++) {
auto q = p[k];
p[k][0] = 0;
for (int i = 1; i <= l; i++) {
mint v = i * q[i];
for (int j = 1; j < i; j++) v -= j * p[k][j] * q[i - j];
p[k][i] = v * inv[i];
}
}
sc.ranked_mobius(p);
return sc.unlift(p);
}
// log(a), [x^0]a=1
// not require inverse of 1,...,sz
template <class mint, int sz = 21>
vector<mint> log_arbitrary(vector<mint> a) {
assert(a[0] == 1);
int l = __builtin_ctz(a.size());
if (l == 0) return {0};
a[0] = 0;
vector<mint> f(l + 1, 0);
f[1] = 1;
for (int i = 2; i <= l; i++) f[i] = f[i - 1] * (1 - i);
return composite_egf<mint, sz>(f, a);
}
// a^m, [x^0]a=1
// require inverse of 1,...,sz
template <class mint, int sz = 21>
vector<mint> pow(vector<mint> a, mint m) {
static SubsetConvolution<mint, sz> sc;
assert(a[0] == 1);
int l = __builtin_ctz(a.size());
if (l == 0) return {1};
vector<mint> inv(l + 1, 1);
rep(i, 1, l + 1) inv[i] = mint(i).inv();
auto p = sc.lift(a);
sc.ranked_zeta(p);
for (int k = 0; k < p.size(); k++) {
auto q = p[k];
p[k][0] = 1;
for (int i = 1; i <= l; i++) {
mint v = 0;
for (int j = 1; j < i; j++) v += (m * j - (i - j)) * p[k][i - j] * q[j];
v *= inv[i];
v += m * p[k][0] * q[i];
p[k][i] = v;
}
}
sc.ranked_mobius(p);
return sc.unlift(p);
}
// a^m, [x^0]a=1
// not require inverse of 1,...,sz
template <class mint, int sz = 21>
vector<mint> pow_arbitrary(vector<mint> a, mint m) {
assert(a[0] == 1);
int l = __builtin_ctz(a.size());
if (l == 0) return {1};
a[0] = 0;
vector<mint> f(l + 1, 0);
f[0] = 1;
for (int i = 1; i <= l; i++) {
f[i] = f[i - 1] * m;
m -= 1;
}
return composite_egf<mint, sz>(f, a);
}
}; // namespace SetPowerSeries
/**
* @brief Polynomial Composite Set Power Series
* @docs docs/set/composite-set-power-series.md
*/
#line 9 "verify/set/UNIT_composite_set_power_series.test.cpp"
SubsetConvolution<mint, 21> sc;
void test_log(vector<mint> a) {
a[0] = 0;
{
auto b = SetPowerSeries::exp(a);
b = SetPowerSeries::log(b);
assert(a == b);
}
a[0] = 1;
{
auto b = SetPowerSeries::log(a);
b = SetPowerSeries::exp(b);
assert(a == b);
}
{
auto b = SetPowerSeries::log(a);
auto c = SetPowerSeries::log_arbitrary(a);
assert(b == c);
}
}
void test_pow(vector<mint> a) {
a[0] = 1;
{
// sqrt
auto b = SetPowerSeries::pow(a, mint(2).inv());
b = sc.multiply(b, b);
assert(a == b);
}
{
mint m = 12345;
auto b = SetPowerSeries::pow(a, m);
auto c = SetPowerSeries::pow_arbitrary(a, m);
assert(b == c);
auto d = SetPowerSeries::log(a);
for (auto& v : d) v *= m;
d = SetPowerSeries::exp(d);
assert(b == d);
}
}
int main() {
rep(k, 0, 10) {
vector<mint> a(1 << k);
iota(ALL(a), 0);
test_log(a);
}
rep(k, 0, 10) {
vector<mint> a(1 << k);
iota(ALL(a), 0);
test_pow(a);
}
int a, b;
in(a, b);
out(a + b);
}
verify/set/UNIT_composite_set_power_series.test.cpp