【多项式模版类】
#include<bits/stdc++.h>
using namespace std;
#define endl '\n'
#define whiteink signed main
#define fi first
#define sc second
#define YES cout<<"YES"<<endl
#define NO cout<<"NO"<<endl
#define Yes cout<<"Yes"<<endl
#define No cout<<"No"<<endl
#define yes cout<<"yes"<<endl
#define no cout<<"no"<<endl
#define pb push_back
#define pq priority_queue
#define all(x) x.begin(),x.end()
using i64=long long;
using i128=__int128;
using u64=unsigned long long;
using u128 = unsigned __int128;
typedef pair<int,int> PII;
typedef pair<i64,i64> P64;
mt19937_64 rng;
//std::random_device rd;
//unsigned int seed = rd();
//rng.seed(seed);
//rng();
template<typename T>
T whink_max(T a,T b){return a>b?a:b;}
template<typename T>
T whink_min(T a,T b){return a<b?a:b;}
template<typename T>
bool cmp(T a,T b){return a>b;}
const int inf_int=0x3f3f3f3f;
const i64 inf_i64=0x3f3f3f3f3f3f3f3f;
//快读快写
inline char nc(){
static char buf[1000000],*p1=buf,*p2=buf;
return p1==p2 && (p2=(p1=buf)+fread(buf,1,1000000,stdin),p1==p2)?EOF:*p1++;
}
template <typename T> void read(T &x){
x=0;
T f=1;
char ch=nc();
while(ch<'0'||ch>'9'){
if(ch=='-') f=-1;
ch=nc();
}
while(ch>='0' && ch<='9'){
x=x*10+ch-'0';
ch=nc();
}
x*=f;
}
template <typename T> void write(T x){
if(x<0){
putchar('-');
x=-x;
}
if(x>9) write(x/10);
putchar(x%10+'0');
}
//快速幂
template<class T>
constexpr T power(T a, unsigned long long b, T res = 1) {
for (; b != 0; b /= 2, a *= a) {
if (b & 1) res *= a;
}
return res;
}
template<unsigned int P>
constexpr unsigned int mulMod(unsigned int a, unsigned int b) {
return (unsigned long long)a * b % P;
}
constexpr long long safeMod(long long x, long long m) {
x %= m;
if (x < 0) x += m;
return x;
}
//扩展欧几里得
constexpr std::pair<long long, long long> invGcd(long long a, long long b) {
a = safeMod(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;
std::swap(s, t);
std::swap(m0, m1);
}
if (m0 < 0) m0 += b / s;
return {s, m0};
}
//模数类
template<unsigned int P>
struct ModIntBase {
public:
unsigned int x;
constexpr ModIntBase() : x(0) {}
template<std::unsigned_integral T>
constexpr ModIntBase(T x_) : x(x_ % P) {}
template<std::signed_integral T>
constexpr ModIntBase(T x_) {
long long v = x_ % (long long)P;
if (v < 0) v += P;
x = (unsigned int)v;
}
constexpr static unsigned int mod() { return P; }
constexpr unsigned int val() const { return x; }
constexpr ModIntBase operator-() const {
ModIntBase res;
res.x = (x == 0 ? 0 : P - x);
return res;
}
constexpr ModIntBase inv() const {
auto v = invGcd(x, P);
return (int)v.second;
}
constexpr ModIntBase &operator*=(const ModIntBase &rhs) & {
x = mulMod<P>(x, rhs.val());
return *this;
}
constexpr ModIntBase &operator+=(const ModIntBase &rhs) & {
x += rhs.val();
if (x >= P) x -= P;
return *this;
}
constexpr ModIntBase &operator-=(const ModIntBase &rhs) & {
if (x < rhs.val()) x += P;
x -= rhs.val();
return *this;
}
constexpr ModIntBase &operator/=(const ModIntBase &rhs) & {
return *this *= rhs.inv();
}
friend constexpr ModIntBase operator*(ModIntBase lhs, const ModIntBase &rhs) { lhs *= rhs; return lhs; }
friend constexpr ModIntBase operator+(ModIntBase lhs, const ModIntBase &rhs) { lhs += rhs; return lhs; }
friend constexpr ModIntBase operator-(ModIntBase lhs, const ModIntBase &rhs) { lhs -= rhs; return lhs; }
friend constexpr ModIntBase operator/(ModIntBase lhs, const ModIntBase &rhs) { lhs /= rhs; return lhs; }
friend constexpr std::istream &operator>>(std::istream &is, ModIntBase &a) {
long long i; is >> i; a = i; return is;
}
friend constexpr std::ostream &operator<<(std::ostream &os, const ModIntBase &a) {
return os << a.val();
}
friend constexpr bool operator==(const ModIntBase &lhs, const ModIntBase &rhs) {
return lhs.val() == rhs.val();
}
friend constexpr std::strong_ordering operator<=>(const ModIntBase &lhs, const ModIntBase &rhs) {
return lhs.val() <=> rhs.val();
}
};
template<int P>
using MInt = ModIntBase<(unsigned int)P>;
using Z = MInt<998244353>;
vector<int> rev;
template<int P> vector<MInt<P>> roots{0, 1};
template<int P>
constexpr MInt<P> findPrimitiveRoot() {
MInt<P> i = 2;
int k = __builtin_ctz(P - 1);
while (true) {
if (power(i, (P - 1) / 2) != 1) break;
i += 1;
}
return power(i, (P - 1) >> k);
}
template<int P> constexpr MInt<P> primitiveRoot = findPrimitiveRoot<P>();
template<> constexpr MInt<998244353> primitiveRoot<998244353>{31};
//dft
template<int P>
constexpr void dft(vector<MInt<P>> &a) {
int n = a.size();
if ((int)rev.size() != n) {
int k = __builtin_ctz(n) - 1;
rev.resize(n);
for (int i = 0; i < n; i++) {
rev[i] = rev[i >> 1] >> 1 | (i & 1) << k;
}
}
for (int i = 0; i < n; i++) {
if (rev[i] < i) swap(a[i], a[rev[i]]);
}
if ((int)roots<P>.size() < n) {
int k = __builtin_ctz(roots<P>.size());
roots<P>.resize(n);
while ((1 << k) < n) {
auto e = power(primitiveRoot<P>, 1 << (__builtin_ctz(P - 1) - k - 1));
for (int i = 1 << (k - 1); i < (1 << k); i++) {
roots<P>[2 * i] = roots<P>[i];
roots<P>[2 * i + 1] = roots<P>[i] * e;
}
k++;
}
}
for (int k = 1; k < n; k *= 2) {
for (int i = 0; i < n; i += 2 * k) {
for (int j = 0; j < k; j++) {
MInt<P> u = a[i + j];
MInt<P> v = a[i + j + k] * roots<P>[k + j];
a[i + j] = u + v;
a[i + j + k] = u - v;
}
}
}
}
//idft
template<int P>
constexpr void idft(vector<MInt<P>> &a) {
int n = a.size();
reverse(a.begin() + 1, a.end());
dft<P>(a);
MInt<P> inv = (1LL - P) / n;
for (int i = 0; i < n; i++) a[i] *= inv;
}
//多项式封装
template<int P = 998244353>
struct Poly : public vector<MInt<P>> {
using Value = MInt<P>;
using vector<Value>::vector;
Poly() : vector<Value>() {}
explicit constexpr Poly(int n) : vector<Value>(n) {}
explicit constexpr Poly(const vector<Value> &a) : vector<Value>(a) {}
constexpr Poly(const initializer_list<Value> &a) : vector<Value>(a) {}
template<class InputIt>
explicit constexpr Poly(InputIt first, InputIt last) : vector<Value>(first, last) {}
template<class F>
explicit constexpr Poly(int n, F f) : vector<Value>(n) {
for (int i = 0; i < n; i++) (*this)[i] = f(i);
}
// ---------- 基础操作 ----------
//将多项式系数整体平移：
// k>0左移，高次补0
// k<0右移，次数降低
constexpr Poly shift(int k) const {
if (k >= 0) {
auto b = *this;
b.insert(b.begin(), k, 0);
return b;
} else if (this->size() <= -k) {
return Poly();
} else {
return Poly(this->begin() + (-k), this->end());
}
}
//截断到(k-1)次
constexpr Poly trunc(int k) const {
Poly f = *this;
f.resize(k);
return f;
}
friend constexpr Poly operator+(const Poly &a, const Poly &b) {
Poly res(max(a.size(), b.size()));
for (int i = 0; i < (int)a.size(); i++) res[i] += a[i];
for (int i = 0; i < (int)b.size(); i++) res[i] += b[i];
return res;
}
friend constexpr Poly operator-(const Poly &a, const Poly &b) {
Poly res(max(a.size(), b.size()));
for (int i = 0; i < (int)a.size(); i++) res[i] += a[i];
for (int i = 0; i < (int)b.size(); i++) res[i] -= b[i];
return res;
}
friend constexpr Poly operator-(const Poly &a) {
vector<Value> res(a.size());
for (int i = 0; i < (int)res.size(); i++) res[i] = -a[i];
return Poly(res);
}
friend constexpr Poly operator*(Poly a, Poly b) {
if (a.size() == 0 || b.size() == 0) return Poly();
if (a.size() < b.size()) swap(a, b);
int n = 1, tot = a.size() + b.size() - 1;
while (n < tot) n *= 2;
if (((P - 1) & (n - 1)) != 0 || b.size() < 128) {
Poly c(a.size() + b.size() - 1);
for (int i = 0; i < (int)a.size(); i++)
for (int j = 0; j < (int)b.size(); j++)
c[i + j] += a[i] * b[j];
return c;
}
a.resize(n); b.resize(n);
dft<P>(a); dft<P>(b);
for (int i = 0; i < n; ++i) a[i] *= b[i];
idft<P>(a);
a.resize(tot);
return a;
}
friend constexpr Poly operator*(Value a, Poly b) { for (auto &v : b) v *= a; return b; }
friend constexpr Poly operator*(Poly a, Value b) { for (auto &v : a) v *= b; return a; }
friend constexpr Poly operator/(Poly a, Value b) { for (auto &v : a) v /= b; return a; }
constexpr Poly &operator+=(Poly b) { return (*this) = (*this) + b; }
constexpr Poly &operator-=(Poly b) { return (*this) = (*this) - b; }
constexpr Poly &operator*=(Poly b) { return (*this) = (*this) * b; }
constexpr Poly &operator*=(Value b) { return (*this) = (*this) * b; }
constexpr Poly &operator/=(Value b) { return (*this) = (*this) / b; }
// ---------- 微积分 ----------
constexpr Poly deriv() const {
if (this->empty()) return Poly();
Poly res(this->size() - 1);
for (int i = 0; i < (int)this->size() - 1; ++i)
res[i] = (i + 1) * (*this)[i + 1];
return res;
}
constexpr Poly integr() const {
Poly res(this->size() + 1);
for (int i = 0; i < (int)this->size(); ++i)
res[i + 1] = (*this)[i] / (i + 1);
return res;
}
// ---------- 多项式高级运算 ----------
constexpr Poly inv(int m) const {
Poly x{(*this)[0].inv()};
int k = 1;
while (k < m) {
k *= 2;
x = (x * (Poly{2} - trunc(k) * x)).trunc(k);
}
return x.trunc(m);
}
constexpr Poly log(int m) const {
return (deriv() * inv(m)).integr().trunc(m);
}
constexpr Poly exp(int m) const {
Poly x{1};
int k = 1;
while (k < m) {
k *= 2;
x = (x * (Poly{1} - x.log(k) + trunc(k))).trunc(k);
}
return x.trunc(m);
}
constexpr Poly pow(int k, int m) const {
int i = 0;
while (i < (int)this->size() && (*this)[i] == 0) i++;
if (i == (int)this->size() || 1LL * i * k >= m) return Poly(m);
Value v = (*this)[i];
auto f = shift(-i) * v.inv();
return (f.log(m - i * k) * k).exp(m - i * k).shift(i * k) * power(v, k);
}
constexpr Poly sqrt(int m) const {
Poly x{1};
int k = 1;
while (k < m) {
k *= 2;
x = (x + (trunc(k) * x.inv(k)).trunc(k)) * MInt<P>(2).inv(); // 乘以 1/2
}
return x.trunc(m);
}
// 中项卷积，用于多项式求值中的多项式取模
constexpr Poly mulT(Poly b) const {
if (b.size() == 0) return Poly();
int n = b.size();
reverse(b.begin(), b.end());
return ((*this) * b).shift(-(n - 1));
}
// 多点求值
constexpr vector<Value> eval(vector<Value> x) const {
if (this->size() == 0) return vector<Value>(x.size(), 0);
int n = max(x.size(), this->size());
vector<Poly> q(4 * n);
vector<Value> ans(x.size());
x.resize(n);
function<void(int, int, int)> build = [&](int p, int l, int r) {
if (r - l == 1) q[p] = Poly{1, -x[l]};
else {
int m = (l + r) / 2;
build(2 * p, l, m);
build(2 * p + 1, m, r);
q[p] = q[2 * p] * q[2 * p + 1];
}
};
build(1, 0, n);
function<void(int, int, int, const Poly &)> work = [&](int p, int l, int r, const Poly &num) {
if (r - l == 1) {
if (l < (int)ans.size()) ans[l] = num[0];
} else {
int m = (l + r) / 2;
work(2 * p, l, m, num.mulT(q[2 * p + 1]).resize(m - l));
work(2 * p + 1, m, r, num.mulT(q[2 * p]).resize(r - m));
}
};
work(1, 0, n, mulT(q[1].inv(n)));
return ans;
}
};
const int N=3e5+10;
int n;
void solve(){
}
whiteink(){
ios::sync_with_stdio(0);
cin.tie(0);
cout.tie(0);
int T=1;
cin>>T;
while(T--) solve();
return 0;
}
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