noya2_Library
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test/fps/Shift_of_Sampling_Points_of_Polynomial.test.cpp
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- Last update: 2025-01-28 23:59:05+09:00
- Problem: https://judge.yosupo.jp/problem/shift_of_sampling_points_of_polynomial
Depends on
- fps/formal_power_series.hpp
- fps/fps_ntt.hpp
- fps/ntt.hpp
- fps/sample_point_shift.hpp
- math/binomial.hpp
- math/prime.hpp
- template/const.hpp
- template/inout_old.hpp
- template/template.hpp
- template/utils.hpp
- utility/modint.hpp
Code
#define PROBLEM "https://judge.yosupo.jp/problem/shift_of_sampling_points_of_polynomial"
#include"../../template/template.hpp"
#include"../../fps/fps_ntt.hpp"
#include"../../fps/sample_point_shift.hpp"
using mint = modint998244353;
using fps = FPS_ntt<mint>;
int main(){
int n, m; in(n,m);
mint c; in(c);
fps y(n); in(y);
out(sample_point_shift(y,c,m));
}#line 1 "test/fps/Shift_of_Sampling_Points_of_Polynomial.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/shift_of_sampling_points_of_polynomial"
#line 2 "template/template.hpp"
using namespace std;
#include<bits/stdc++.h>
#line 1 "template/inout_old.hpp"
namespace noya2 {
template <typename T, typename U>
ostream &operator<<(ostream &os, const pair<T, U> &p){
os << p.first << " " << p.second;
return os;
}
template <typename T, typename U>
istream &operator>>(istream &is, pair<T, U> &p){
is >> p.first >> p.second;
return is;
}
template <typename T>
ostream &operator<<(ostream &os, const vector<T> &v){
int s = (int)v.size();
for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i];
return os;
}
template <typename T>
istream &operator>>(istream &is, vector<T> &v){
for (auto &x : v) is >> x;
return is;
}
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...);
}
template<typename T>
void out(const vector<vector<T>> &vv){
int s = (int)vv.size();
for (int i = 0; i < s; i++) out(vv[i]);
}
struct IoSetup {
IoSetup(){
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(15);
cerr << fixed << setprecision(7);
}
} iosetup_noya2;
} // namespace noya2
#line 1 "template/const.hpp"
namespace noya2{
const int iinf = 1'000'000'007;
const long long linf = 2'000'000'000'000'000'000LL;
const long long mod998 = 998244353;
const long long mod107 = 1000000007;
const long double pi = 3.14159265358979323;
const vector<int> dx = {0,1,0,-1,1,1,-1,-1};
const vector<int> dy = {1,0,-1,0,1,-1,-1,1};
const string ALP = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
const string alp = "abcdefghijklmnopqrstuvwxyz";
const string NUM = "0123456789";
void yes(){ cout << "Yes\n"; }
void no(){ cout << "No\n"; }
void YES(){ cout << "YES\n"; }
void NO(){ cout << "NO\n"; }
void yn(bool t){ t ? yes() : no(); }
void YN(bool t){ t ? YES() : NO(); }
} // namespace noya2
#line 2 "template/utils.hpp"
#line 6 "template/utils.hpp"
namespace noya2{
unsigned long long inner_binary_gcd(unsigned long long a, unsigned long long b){
if (a == 0 || b == 0) return a + b;
int n = __builtin_ctzll(a); a >>= n;
int m = __builtin_ctzll(b); b >>= m;
while (a != b) {
int mm = __builtin_ctzll(a - b);
bool f = a > b;
unsigned long long c = f ? a : b;
b = f ? b : a;
a = (c - b) >> mm;
}
return a << std::min(n, m);
}
template<typename T> T gcd_fast(T a, T b){ return static_cast<T>(inner_binary_gcd(std::abs(a),std::abs(b))); }
long long sqrt_fast(long long n) {
if (n <= 0) return 0;
long long x = sqrt(n);
while ((x + 1) * (x + 1) <= n) x++;
while (x * x > n) x--;
return x;
}
template<typename T> T floor_div(const T n, const T d) {
assert(d != 0);
return n / d - static_cast<T>((n ^ d) < 0 && n % d != 0);
}
template<typename T> T ceil_div(const T n, const T d) {
assert(d != 0);
return n / d + static_cast<T>((n ^ d) >= 0 && n % d != 0);
}
template<typename T> void uniq(std::vector<T> &v){
std::sort(v.begin(),v.end());
v.erase(unique(v.begin(),v.end()),v.end());
}
template <typename T, typename U> inline bool chmin(T &x, U y) { return (y < x) ? (x = y, true) : false; }
template <typename T, typename U> inline bool chmax(T &x, U y) { return (x < y) ? (x = y, true) : false; }
template<typename T> inline bool range(T l, T x, T r){ return l <= x && x < r; }
} // namespace noya2
#line 8 "template/template.hpp"
#define rep(i,n) for (int i = 0; i < (int)(n); i++)
#define repp(i,m,n) for (int i = (m); i < (int)(n); i++)
#define reb(i,n) for (int i = (int)(n-1); i >= 0; i--)
#define all(v) (v).begin(),(v).end()
using ll = long long;
using ld = long double;
using uint = unsigned int;
using ull = unsigned long long;
using pii = pair<int,int>;
using pll = pair<ll,ll>;
using pil = pair<int,ll>;
using pli = pair<ll,int>;
namespace noya2{
/* ~ (. _________ . /) */
}
using namespace noya2;
#line 2 "fps/fps_ntt.hpp"
#line 2 "fps/formal_power_series.hpp"
#line 4 "fps/formal_power_series.hpp"
namespace noya2{
template<typename T>
concept Field = requires (T a, T b){
a + b; a - b; a / b; a * b;
T(0); T(1);
};
template<class Info>
concept Fps_Info = requires {
typename Info::value_type;
requires Field<typename Info::value_type>;
{Info::multiply(declval<vector<typename Info::value_type>>(),declval<vector<typename Info::value_type>>())} -> convertible_to<vector<typename Info::value_type>>;
{Info::inv(declval<vector<typename Info::value_type>>(),declval<int>())} -> convertible_to<vector<typename Info::value_type>>;
{Info::integral(declval<vector<typename Info::value_type>>())} -> convertible_to<vector<typename Info::value_type>>;
};
template<Fps_Info Info>
struct FormalPowerSeries : vector<typename Info::value_type> {
using T = typename Info::value_type;
using vector<T>::vector;
using vector<T>::operator=;
using FPS = FormalPowerSeries;
FormalPowerSeries (const vector<T> &init_ = {}){ (*this) = init_; }
void shrink(){ while (!(*this).empty() && (*this).back() == T(0)) (*this).pop_back(); }
FPS &operator+=(const T &r){
if ((*this).empty()) (*this).resize(1);
(*this)[0] += r;
return *this;
}
FPS &operator-=(const T &r){
if ((*this).empty()) (*this).resize(1);
(*this)[0] -= r;
return *this;
}
FPS &operator*=(const T &r){
for (auto &x : *this) x *= r;
return *this;
}
FPS &operator/=(const T &r){
(*this) *= T(1)/r;
return *this;
}
FPS &operator<<=(const int &d){
(*this).insert((*this).begin(),d,T(0));
return *this;
}
FPS &operator>>=(const int &d){
if ((int)(*this).size() <= d) (*this).clear();
else (*this).erase((*this).begin(),(*this).begin()+d);
return *this;
}
FPS &operator+=(const FPS &r){
if ((*this).size() < r.size()) (*this).resize(r.size());
for (int i = 0; i < (int)(r.size()); i++) (*this)[i] += r[i];
return *this;
}
FPS &operator-=(const FPS &r){
if ((*this).size() < r.size()) (*this).resize(r.size());
for (int i = 0; i < (int)(r.size()); i++) (*this)[i] -= r[i];
return *this;
}
FPS &operator*=(const FPS &r){
if ((*this).empty() || r.empty()){
(*this).clear();
return *this;
}
(*this) = Info::multiply(*this,r);
return *this;
}
FPS operator+(const T &r) const { return FPS(*this) += r; }
FPS operator-(const T &r) const { return FPS(*this) -= r; }
FPS operator*(const T &r) const { return FPS(*this) *= r; }
FPS operator/(const T &r) const { return FPS(*this) /= r; }
FPS operator<<(const int &d) const { return FPS(*this) <<= d; }
FPS operator>>(const int &d) const { return FPS(*this) >>= d; }
FPS operator+(const FPS &r) const { return FPS(*this) += r; }
FPS operator-(const FPS &r) const { return FPS(*this) -= r; }
FPS operator*(const FPS &r) const { return FPS(*this) *= r; }
FPS operator+() const { return *this; }
FPS operator-() const {
FPS res(*this);
for (auto &x : res) x = -x;
return res;
}
T eval(const T &x) const {
T res = T(0), w = T(1);
for (auto &e : *this) res += e * w, w *= x;
return res;
}
static FPS dot(const FPS &lhs, const FPS &rhs){
FPS res(min(lhs.size(),rhs.size()));
for (int i = 0; i < (int)res.size(); i++) res[i] = lhs[i] * rhs[i];
return res;
}
FPS pre(int siz) const {
FPS ret((*this).begin(), (*this).begin() + min((int)this->size(), siz));
if ((int)ret.size() < siz) ret.resize(siz);
return ret;
}
FPS rev() const {
FPS ret(*this);
reverse(ret.begin(), ret.end());
return ret;
}
FPS diff() const {
const int n = (int)this->size();
FPS ret(max(0, n - 1));
T one(1), coeff(1);
for (int i = 1; i < n; i++) {
ret[i - 1] = (*this)[i] * coeff;
coeff += one;
}
return ret;
}
FPS integral() const {
FPS ret = Info::integral(*this);
return ret;
}
FPS inv(int d = -1) const {
FPS ret = Info::inv(*this,d);
return ret;
}
FPS exp(int d = -1) const {
const int n = (*this).size();
if (d == -1) d = n;
FPS f = {T(1)+(*this)[0],(*this)[1]}, res = {1,(n > 1 ? (*this)[1] : 0)};
for (int sz = 2; sz < d; sz <<= 1){
f.insert(f.end(),(*this).begin()+min(n,sz),(*this).begin()+min(n,sz*2));
if ((int)f.size() < sz*2) f.resize(sz*2);
res = res * (f - res.log(2*sz));
res.resize(sz*2);
}
res.resize(d);
return res;
}
FPS log(int d = -1) const {
assert(!(*this).empty() && (*this)[0] == T(1));
if (d == -1) d = (*this).size();
return (this->diff() * this->inv(d)).pre(d - 1).integral();
}
};
} // namespace noya2
#line 2 "math/binomial.hpp"
#line 4 "math/binomial.hpp"
namespace noya2 {
template<typename mint>
struct binomial {
binomial(int len = 300000){ extend(len); }
static mint fact(int n){
if (n < 0) return 0;
while (n >= (int)_fact.size()) extend();
return _fact[n];
}
static mint ifact(int n){
if (n < 0) return 0;
while (n >= (int)_fact.size()) extend();
return _ifact[n];
}
static mint inv(int n){
return ifact(n) * fact(n-1);
}
static mint C(int n, int r){
if (!(0 <= r && r <= n)) return 0;
return fact(n) * ifact(r) * ifact(n-r);
}
static mint P(int n, int r){
if (!(0 <= r && r <= n)) return 0;
return fact(n) * ifact(n-r);
}
static mint catalan(int n){
return C(n * 2, n) * inv(n + 1);
}
inline mint operator()(int n, int r) { return C(n, r); }
template<class... Cnts>
static mint M(const Cnts&... cnts){
return multinomial(0,1,cnts...);
}
static void initialize(int len = 2){
_fact.clear();
_ifact.clear();
extend(len);
}
private:
static mint multinomial(const int& sum, const mint& div_prod){
if (sum < 0) return 0;
return fact(sum) * div_prod;
}
template<class... Tail>
static mint multinomial(const int& sum, const mint& div_prod, const int& n1, const Tail&... tail){
if (n1 < 0) return 0;
return multinomial(sum+n1,div_prod*ifact(n1),tail...);
}
static inline std::vector<mint> _fact, _ifact;
static void extend(int len = -1){
if (_fact.empty()){
_fact = _ifact = {1,1};
}
int siz = _fact.size();
if (len == -1) len = siz * 2;
len = (int)min<long long>(len, mint::mod() - 1);
if (len < siz) return ;
_fact.resize(len+1), _ifact.resize(len+1);
for (int i = siz; i <= len; i++) _fact[i] = _fact[i-1] * i;
_ifact[len] = _fact[len].inv();
for (int i = len; i > siz; i--) _ifact[i-1] = _ifact[i] * i;
}
};
} // namespace noya2
#line 2 "fps/ntt.hpp"
#line 2 "utility/modint.hpp"
#line 4 "utility/modint.hpp"
#line 2 "math/prime.hpp"
#line 4 "math/prime.hpp"
namespace noya2 {
constexpr long long safe_mod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
return x;
}
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
if (m == 1) return 0;
unsigned int _m = (unsigned int)(m);
unsigned long long r = 1;
unsigned long long y = safe_mod(x, m);
while (n) {
if (n & 1) r = (r * y) % _m;
y = (y * y) % _m;
n >>= 1;
}
return r;
}
constexpr bool is_prime_constexpr(int n) {
if (n <= 1) return false;
if (n == 2 || n == 7 || n == 61) return true;
if (n % 2 == 0) return false;
long long d = n - 1;
while (d % 2 == 0) d /= 2;
constexpr long long bases[3] = {2, 7, 61};
for (long long a : bases) {
long long t = d;
long long y = pow_mod_constexpr(a, t, n);
while (t != n - 1 && y != 1 && y != n - 1) {
y = y * y % n;
t <<= 1;
}
if (y != n - 1 && t % 2 == 0) {
return false;
}
}
return true;
}
template <int n> constexpr bool is_prime_flag = is_prime_constexpr(n);
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
a = safe_mod(a, b);
if (a == 0) return {b, 0};
long long s = b, t = a;
long long m0 = 0, m1 = 1;
while (t) {
long long u = s / t;
s -= t * u;
m0 -= m1 * u;
auto tmp = s;
s = t;
t = tmp;
tmp = m0;
m0 = m1;
m1 = tmp;
}
if (m0 < 0) m0 += b / s;
return {s, m0};
}
constexpr int primitive_root_constexpr(int m) {
if (m == 2) return 1;
if (m == 167772161) return 3;
if (m == 469762049) return 3;
if (m == 754974721) return 11;
if (m == 998244353) return 3;
int divs[20] = {};
divs[0] = 2;
int cnt = 1;
int x = (m - 1) / 2;
while (x % 2 == 0) x /= 2;
for (int i = 3; (long long)(i)*i <= x; i += 2) {
if (x % i == 0) {
divs[cnt++] = i;
while (x % i == 0) {
x /= i;
}
}
}
if (x > 1) {
divs[cnt++] = x;
}
for (int g = 2;; g++) {
bool ok = true;
for (int i = 0; i < cnt; i++) {
if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
ok = false;
break;
}
}
if (ok) return g;
}
}
template <int m> constexpr int primitive_root_flag = primitive_root_constexpr(m);
} // namespace noya2
#line 6 "utility/modint.hpp"
namespace noya2{
struct barrett {
unsigned int _m;
unsigned long long im;
explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}
unsigned int umod() const { return _m; }
unsigned int mul(unsigned int a, unsigned int b) const {
unsigned long long z = a;
z *= b;
unsigned long long x = (unsigned long long)((__uint128_t(z) * im) >> 64);
unsigned int v = (unsigned int)(z - x * _m);
if (_m <= v) v += _m;
return v;
}
};
template <int m>
struct static_modint {
using mint = static_modint;
public:
static constexpr int mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
constexpr static_modint() : _v(0) {}
template<std::signed_integral T>
constexpr static_modint(T v){
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
template<std::unsigned_integral T>
constexpr static_modint(T v){
_v = (unsigned int)(v % umod());
}
constexpr 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;
}
constexpr mint& operator+=(const mint& rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
constexpr mint& operator-=(const mint& rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
constexpr mint& operator*=(const mint& rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (uint)(z % umod());
return *this;
}
constexpr mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }
constexpr mint operator+() const { return *this; }
constexpr mint operator-() const { return mint() - *this; }
constexpr 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;
}
constexpr mint inv() const {
if (prime) {
assert(_v);
return pow(umod() - 2);
} else {
auto eg = inv_gcd(_v, m);
assert(eg.first == 1);
return eg.second;
}
}
friend constexpr mint operator+(const mint& lhs, const mint& rhs) {
return mint(lhs) += rhs;
}
friend constexpr mint operator-(const mint& lhs, const mint& rhs) {
return mint(lhs) -= rhs;
}
friend constexpr mint operator*(const mint& lhs, const mint& rhs) {
return mint(lhs) *= rhs;
}
friend constexpr mint operator/(const mint& lhs, const mint& rhs) {
return mint(lhs) /= rhs;
}
friend constexpr bool operator==(const mint& lhs, const mint& rhs) {
return lhs._v == rhs._v;
}
friend constexpr bool operator!=(const mint& lhs, const mint& rhs) {
return lhs._v != rhs._v;
}
friend std::ostream &operator<<(std::ostream &os, const mint& p) {
return os << p.val();
}
friend std::istream &operator>>(std::istream &is, mint &a) {
long long t; is >> t;
a = mint(t);
return (is);
}
private:
unsigned int _v;
static constexpr unsigned int umod() { return m; }
static constexpr bool prime = is_prime_flag<m>;
};
template <int id> struct dynamic_modint {
using mint = dynamic_modint;
public:
static int mod() { return (int)(bt.umod()); }
static void set_mod(int m) {
assert(1 <= m);
bt = barrett(m);
}
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
dynamic_modint() : _v(0) {}
template<std::signed_integral T>
dynamic_modint(T v){
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
template<std::unsigned_integral T>
dynamic_modint(T v){
_v = (unsigned int)(v % umod());
}
uint 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 += mod() - rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint& operator*=(const mint& rhs) {
_v = bt.mul(_v, rhs._v);
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 {
auto eg = noya2::inv_gcd(_v, mod());
assert(eg.first == 1);
return eg.second;
}
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 std::ostream &operator<<(std::ostream &os, const mint& p) {
return os << p.val();
}
friend std::istream &operator>>(std::istream &is, mint &a) {
long long t; is >> t;
a = mint(t);
return (is);
}
private:
unsigned int _v;
static barrett bt;
static unsigned int umod() { return bt.umod(); }
};
template <int id> noya2::barrett dynamic_modint<id>::bt(998244353);
using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;
template<typename T>
concept Modint = requires (T &a){
T::mod();
a.inv();
a.val();
a.pow(declval<int>());
};
} // namespace noya2
#line 5 "fps/ntt.hpp"
namespace noya2{
template<Modint mint>
struct NTT {
static constexpr uint mod = mint::mod();
static constexpr ull mod2 = (ull)mod * mod;
static constexpr uint pr = primitive_root_constexpr(mod);
static constexpr int level = countr_zero(mod-1);
mint wp[level+1], wm[level+1];
void set_ws(){
mint r = mint(pr).pow((mod-1) >> level);
wp[level] = r, wm[level] = r.inv();
for (int i = level-1; i >= 0; i--){
wp[i] = wp[i+1] * wp[i+1];
wm[i] = wm[i+1] * wm[i+1];
}
}
NTT () { set_ws(); }
void fft4(vector<mint> &a, int k, int s = 0){
uint im = wm[2].val();
uint n = 1<<k;
uint len = n;
int l = k;
while (len > 1){
if (l == 1){
for (int i = 0; i < (1<<(k-1)); i++){
int i0 = s + i*2, i1 = i0+1;
a[i0] += a[i1];
a[i1] = a[i0] - a[i1] * 2;
}
len >>= 1;
l -= 1;
}
else {
int len4 = len/4;
int nlen = n/len;
ull r1 = 1, r2 = 1, r3 = 1, imr1 = im, imr3 = im;
for (int i = 0; i < len4; i++){
int offset = 0;
for (int j = 0; j < nlen; j++){
int i0 = s + i + offset, i1 = i0 + len4, i2 = i1 + len4, i3 = i2 + len4;
uint a0 = a[i0].val();
uint a1 = a[i1].val();
uint a2 = a[i2].val();
uint a3 = a[i3].val();
uint a0p2 = a0 + a2;
uint a1p3 = a1 + a3;
ull b0m2 = (a0 + mod - a2) * r1;
ull b1m3 = (a1 + mod - a3) * imr1;
ull c0m2 = (a0 + mod - a2) * r3;
ull c1m3 = (a1 + mod - a3) * imr3;
a[i0] = a0p2 + a1p3;
a[i1] = b0m2 + b1m3;
a[i2] = (a0p2 + mod*2 - a1p3) * r2;
a[i3] = c0m2 + mod2*2 - c1m3;
offset += len;
}
r1 = r1 * wm[l].val() % mod;
r2 = r1 * r1 % mod;
r3 = r1 * r2 % mod;
imr1 = im * r1 % mod;
imr3 = im * r3 % mod;
}
len >>= 2;
l -= 2;
}
}
}
void ifft4(vector<mint> &a, int k, int s = 0){
uint im = wp[2].val();
uint n = 1<<k;
uint len = (k & 1 ? 2 : 4);
int l = (k & 1 ? 1 : 2);
while (len <= n){
if (l == 1){
for (int i = 0; i < (1<<(k-1)); i++){
int i0 = s + i*2, i1 = i0+1;
a[i0] += a[i1];
a[i1] = a[i0] - a[i1] * 2;
}
len <<= 2;
l += 2;
}
else {
int len4 = len/4;
int nlen = n/len;
ull r1 = 1, r2 = 1, r3 = 1, imr1 = im, imr3 = im;
for (int i = 0; i < len4; i++){
int offset = 0;
for (int j = 0; j < nlen; j++){
int i0 = s + i + offset, i1 = i0 + len4, i2 = i1 + len4, i3 = i2 + len4;
ull a0 = a[i0].val();
ull a1 = a[i1].val() * r1;
ull a2 = a[i2].val() * r2;
ull a3 = a[i3].val() * r3;
ull b1 = a[i1].val() * imr1;
ull b3 = a[i3].val() * imr3;
ull a0p2 = a0 + a2;
ull a1p3 = a1 + a3;
ull a0m2 = a0 + mod2 - a2;
ull b1m3 = b1 + mod2 - b3;
a[i0] = a0p2 + a1p3;
a[i1] = a0m2 + b1m3;
a[i2] = a0p2 + mod2*2 - a1p3;
a[i3] = a0m2 + mod2*2 - b1m3;
offset += len;
}
r1 = r1 * wp[l].val() % mod;
r2 = r1 * r1 % mod;
r3 = r1 * r2 % mod;
imr1 = im * r1 % mod;
imr3 = im * r3 % mod;
}
len <<= 2;
l += 2;
}
}
}
void ntt(vector<mint> &a) {
if ((int)a.size() <= 1) return;
assert(has_single_bit(a.size()));
fft4(a, countr_zero(a.size()));
}
void intt(vector<mint> &a, bool stop = false) {
if ((int)a.size() <= 1) return;
assert(has_single_bit(a.size()));
ifft4(a, countr_zero(a.size()));
if (stop) return ;
mint iv = mint(a.size()).inv();
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;
set_ws();
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).inv();
for (int i = 0; i < l; ++i) s[i] *= invm;
return s;
}
};
} // namespace noya2
#line 6 "fps/fps_ntt.hpp"
namespace noya2{
template<typename T>
struct fps_ntt{
using value_type = T;
static NTT<T> ntt;
static vector<T> multiply(const vector<T> &a, const vector<T> &b){
return ntt.multiply(a,b);
}
static vector<T> inv(const vector<T> &a, int d = -1){
const int n = a.size();
if (d == -1) d = n;
vector<T> res = {a[0].inv()};
for (int siz = 1; siz < d; siz <<= 1){
vector<T> f(a.begin(),a.begin()+min(n,siz*2)), g(res);
f.resize(siz*2), g.resize(siz*2);
ntt.ntt(f), ntt.ntt(g);
for (int i = 0; i < siz*2; i++) f[i] *= g[i];
ntt.intt(f,true);
f.erase(f.begin(),f.begin()+siz);
f.resize(siz*2);
ntt.ntt(f);
for (int i = 0; i < siz*2; i++) f[i] *= g[i];
ntt.intt(f,true);
T siz2_inv = T(siz*2).inv(); siz2_inv *= -siz2_inv;
for (int i = 0; i < siz; i++) f[i] *= siz2_inv;
res.insert(res.end(),f.begin(),f.begin()+siz);
}
res.resize(d);
return res;
}
static binomial<T> bnm;
static vector<T> integral(const vector<T> &a){
const int n = a.size();
vector<T> res(n+1);
for (int i = 1; i <= n; i++) res[i] = a[i-1] * bnm.inv(i);
return res;
}
};
template<typename T> NTT<T> fps_ntt<T>::ntt;
template<typename T> using FPS_ntt = FormalPowerSeries<fps_ntt<T>>;
} // namespace noya2
#line 2 "fps/sample_point_shift.hpp"
#line 7 "fps/sample_point_shift.hpp"
namespace noya2{
template<Fps_Info Info>
requires Modint<typename Info::value_type>
FormalPowerSeries<Info> sample_point_shift(FormalPowerSeries<Info> y, typename Info::value_type t, int m){
using fps = FormalPowerSeries<Info>;
using mint = typename Info::value_type;
ll T = t.val();
int k = (int)(y.size()) - 1;
if (T <= k){
fps ret(m);
int ptr = 0;
for (ll i = T; i <= k && ptr < m; i++){
ret[ptr++] = y[i];
}
if (k+1 < T+m){
auto suf = sample_point_shift(y,k+1,m-ptr);
for (int i = k+1; i < T+m; i++){
ret[ptr++] = suf[i-(k+1)];
}
}
return ret;
}
if (T+m > mint::mod()){
auto pref = sample_point_shift(y,T,mint::mod()-T);
auto suf = sample_point_shift(y,0,m-(int)(pref.size()));
copy(suf.begin(),suf.end(),back_inserter(pref));
return pref;
}
binomial<mint> bnm;
fps d(k+1);
for (int i = 0; i <= k; i++){
d[i] = bnm.ifact(i) * bnm.ifact(k-i) * y[i];
if ((k-i)&1) d[i] = -d[i];
}
vector<mint> fact(m+k+1); fact[0] = 1;
for (int i = 0; i < m+k; i++) fact[i+1] = fact[i] * (T-k+i);
fps h(m+k); h[m+k-1] = fact[m+k].inv();
for (int i = m+k-1; i >= 1; i--) h[i-1] = h[i] * (T-k+i);
for (int i = 0; i < m+k; i++) h[i] *= fact[i];
auto dh = d * h;
fps ret(m);
mint cur = T;
for (int i = 1; i <= k; i++) cur *= T-i;
for (int i = 0; i < m; i++){
ret[i] = cur * dh[k+i];
cur *= T+i+1;
cur *= h[i];
}
return ret;
}
} // namespace noya2
#line 6 "test/fps/Shift_of_Sampling_Points_of_Polynomial.test.cpp"
using mint = modint998244353;
using fps = FPS_ntt<mint>;
int main(){
int n, m; in(n,m);
mint c; in(c);
fps y(n); in(y);
out(sample_point_shift(y,c,m));
}