結果
| 問題 |
No.2166 Paint and Fill
|
| コンテスト | |
| ユーザー |
 hos.lyric
|
| 提出日時 | 2022-12-18 07:00:06 |
| 言語 | C++17 (gcc 13.3.0 + boost 1.87.0) |
| 結果 |
TLE
|
| 実行時間 | - |
| コード長 | 64,369 bytes |
| コンパイル時間 | 4,514 ms |
| コンパイル使用メモリ | 332,224 KB |
| 最終ジャッジ日時 | 2025-02-09 15:42:24 |
|
ジャッジサーバーID （参考情報） |
judge4 / judge3 |
（要ログイン）
| ファイルパターン | 結果 |
|---|---|
| sample | AC * 2 |
| other | AC * 30 TLE * 8 |
コンパイルメッセージ
template/template.hpp: In function ‘int nyaan::main()’: template/template.hpp:74:30: warning: no return statement in function returning non-void [-Wreturn-type] verify/verify-unit-test/polynomial-matrix-prod.test.cpp: In function ‘int main()’: verify/verify-unit-test/polynomial-matrix-prod.test.cpp:221:12: warning: ignoring return value of ‘int scanf(const char*, ...)’ declared with attribute ‘warn_unused_result’ [-Wunused-result]
ソースコード
// TODO: my own library
// Nyaan orz
// used https://nyaannyaan.github.io/library/verify/verify-unit-test/polynomial-matrix-prod.test.cpp
#include <cassert>
#include <cmath>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <algorithm>
#include <bitset>
#include <complex>
#include <deque>
#include <functional>
#include <iostream>
#include <map>
#include <numeric>
#include <queue>
#include <set>
#include <sstream>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
using namespace std;
using Int = long long;
template <class T1, class T2> ostream &operator<<(ostream &os, const pair<T1, T2> &a) { return os << "(" << a.first << ", " << a.second << ")"; };
template <class T> ostream &operator<<(ostream &os, const vector<T> &as) { const int sz = as.size(); os << "["; for (int i = 0; i < sz; ++i) { if (i >= 256) { os << ", ..."; break; } if (i > 0) { os << ", "; } os << as[i]; } return os << "]"; }
template <class T> void pv(T a, T b) { for (T i = a; i != b; ++i) cerr << *i << " "; cerr << endl; }
template <class T> bool chmin(T &t, const T &f) { if (t > f) { t = f; return true; } return false; }
template <class T> bool chmax(T &t, const T &f) { if (t < f) { t = f; return true; } return false; }
////////////////////////////////////////////////////////////////////////////////
template <unsigned M_> struct ModInt {
static constexpr unsigned M = M_;
unsigned x;
constexpr ModInt() : x(0U) {}
constexpr ModInt(unsigned x_) : x(x_ % M) {}
constexpr ModInt(unsigned long long x_) : x(x_ % M) {}
constexpr ModInt(int x_) : x(((x_ %= static_cast<int>(M)) < 0) ? (x_ + static_cast<int>(M)) : x_) {}
constexpr ModInt(long long x_) : x(((x_ %= static_cast<long long>(M)) < 0) ? (x_ + static_cast<long long>(M)) : x_) {}
ModInt &operator+=(const ModInt &a) { x = ((x += a.x) >= M) ? (x - M) : x; return *this; }
ModInt &operator-=(const ModInt &a) { x = ((x -= a.x) >= M) ? (x + M) : x; return *this; }
ModInt &operator*=(const ModInt &a) { x = (static_cast<unsigned long long>(x) * a.x) % M; return *this; }
ModInt &operator/=(const ModInt &a) { return (*this *= a.inv()); }
ModInt pow(long long e) const {
if (e < 0) return inv().pow(-e);
ModInt a = *this, b = 1U; for (; e; e >>= 1) { if (e & 1) b *= a; a *= a; } return b;
}
ModInt inv() const {
unsigned a = M, b = x; int y = 0, z = 1;
for (; b; ) { const unsigned q = a / b; const unsigned c = a - q * b; a = b; b = c; const int w = y - static_cast<int>(q) * z; y = z; z = w; }
assert(a == 1U); return ModInt(y);
}
ModInt operator+() const { return *this; }
ModInt operator-() const { ModInt a; a.x = x ? (M - x) : 0U; return a; }
ModInt operator+(const ModInt &a) const { return (ModInt(*this) += a); }
ModInt operator-(const ModInt &a) const { return (ModInt(*this) -= a); }
ModInt operator*(const ModInt &a) const { return (ModInt(*this) *= a); }
ModInt operator/(const ModInt &a) const { return (ModInt(*this) /= a); }
template <class T> friend ModInt operator+(T a, const ModInt &b) { return (ModInt(a) += b); }
template <class T> friend ModInt operator-(T a, const ModInt &b) { return (ModInt(a) -= b); }
template <class T> friend ModInt operator*(T a, const ModInt &b) { return (ModInt(a) *= b); }
template <class T> friend ModInt operator/(T a, const ModInt &b) { return (ModInt(a) /= b); }
explicit operator bool() const { return x; }
bool operator==(const ModInt &a) const { return (x == a.x); }
bool operator!=(const ModInt &a) const { return (x != a.x); }
friend std::ostream &operator<<(std::ostream &os, const ModInt &a) { return os << a.x; }
};
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
constexpr unsigned MO = 998244353U;
constexpr unsigned MO2 = 2U * MO;
constexpr int FFT_MAX = 23;
using Mint = ModInt<MO>;
constexpr Mint FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 911660635U, 372528824U, 929031873U, 452798380U, 922799308U, 781712469U, 476477967U, 166035806U, 258648936U, 584193783U, 63912897U, 350007156U, 666702199U, 968855178U, 629671588U, 24514907U, 996173970U, 363395222U, 565042129U, 733596141U, 267099868U, 15311432U};
constexpr Mint INV_FFT_ROOTS[FFT_MAX + 1] = {1U, 998244352U, 86583718U, 509520358U, 337190230U, 87557064U, 609441965U, 135236158U, 304459705U, 685443576U, 381598368U, 335559352U, 129292727U, 358024708U, 814576206U, 708402881U, 283043518U, 3707709U, 121392023U, 704923114U, 950391366U, 428961804U, 382752275U, 469870224U};
constexpr Mint FFT_RATIOS[FFT_MAX] = {911660635U, 509520358U, 369330050U, 332049552U, 983190778U, 123842337U, 238493703U, 975955924U, 603855026U, 856644456U, 131300601U, 842657263U, 730768835U, 942482514U, 806263778U, 151565301U, 510815449U, 503497456U, 743006876U, 741047443U, 56250497U, 867605899U};
constexpr Mint INV_FFT_RATIOS[FFT_MAX] = {86583718U, 372528824U, 373294451U, 645684063U, 112220581U, 692852209U, 155456985U, 797128860U, 90816748U, 860285882U, 927414960U, 354738543U, 109331171U, 293255632U, 535113200U, 308540755U, 121186627U, 608385704U, 438932459U, 359477183U, 824071951U, 103369235U};
// as[rev(i)] <- \sum_j \zeta^(ij) as[j]
void fft(Mint *as, int n) {
assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << FFT_MAX);
int m = n;
if (m >>= 1) {
for (int i = 0; i < m; ++i) {
const unsigned x = as[i + m].x; // < MO
as[i + m].x = as[i].x + MO - x; // < 2 MO
as[i].x += x; // < 2 MO
}
}
if (m >>= 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + m; ++i) {
const unsigned x = (prod * as[i + m]).x; // < MO
as[i + m].x = as[i].x + MO - x; // < 3 MO
as[i].x += x; // < 3 MO
}
prod *= FFT_RATIOS[__builtin_ctz(++h)];
}
}
for (; m; ) {
if (m >>= 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + m; ++i) {
const unsigned x = (prod * as[i + m]).x; // < MO
as[i + m].x = as[i].x + MO - x; // < 4 MO
as[i].x += x; // < 4 MO
}
prod *= FFT_RATIOS[__builtin_ctz(++h)];
}
}
if (m >>= 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + m; ++i) {
const unsigned x = (prod * as[i + m]).x; // < MO
as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x; // < 2 MO
as[i + m].x = as[i].x + MO - x; // < 3 MO
as[i].x += x; // < 3 MO
}
prod *= FFT_RATIOS[__builtin_ctz(++h)];
}
}
}
for (int i = 0; i < n; ++i) {
as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x; // < 2 MO
as[i].x = (as[i].x >= MO) ? (as[i].x - MO) : as[i].x; // < MO
}
}
// as[i] <- (1/n) \sum_j \zeta^(-ij) as[rev(j)]
void invFft(Mint *as, int n) {
assert(!(n & (n - 1))); assert(1 <= n); assert(n <= 1 << FFT_MAX);
int m = 1;
if (m < n >> 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + m; ++i) {
const unsigned long long y = as[i].x + MO - as[i + m].x; // < 2 MO
as[i].x += as[i + m].x; // < 2 MO
as[i + m].x = (prod.x * y) % MO; // < MO
}
prod *= INV_FFT_RATIOS[__builtin_ctz(++h)];
}
m <<= 1;
}
for (; m < n >> 1; m <<= 1) {
Mint prod = 1U;
for (int h = 0, i0 = 0; i0 < n; i0 += (m << 1)) {
for (int i = i0; i < i0 + (m >> 1); ++i) {
const unsigned long long y = as[i].x + MO2 - as[i + m].x; // < 4 MO
as[i].x += as[i + m].x; // < 4 MO
as[i].x = (as[i].x >= MO2) ? (as[i].x - MO2) : as[i].x; // < 2 MO
as[i + m].x = (prod.x * y) % MO; // < MO
}
for (int i = i0 + (m >> 1); i < i0 + m; ++i) {
const unsigned long long y = as[i].x + MO - as[i + m].x; // < 2 MO
as[i].x += as[i + m].x; // < 2 MO
as[i + m].x = (prod.x * y) % MO; // < MO
}
prod *= INV_FFT_RATIOS[__builtin_ctz(++h)];
}
}
if (m < n) {
for (int i = 0; i < m; ++i) {
const unsigned y = as[i].x + MO2 - as[i + m].x; // < 4 MO
as[i].x += as[i + m].x; // < 4 MO
as[i + m].x = y; // < 4 MO
}
}
const Mint invN = Mint(n).inv();
for (int i = 0; i < n; ++i) {
as[i] *= invN;
}
}
void fft(vector<Mint> &as) {
fft(as.data(), as.size());
}
void invFft(vector<Mint> &as) {
invFft(as.data(), as.size());
}
vector<Mint> convolve(vector<Mint> as, vector<Mint> bs) {
if (as.empty() || bs.empty()) return {};
const int len = as.size() + bs.size() - 1;
int n = 1;
for (; n < len; n <<= 1) {}
as.resize(n); fft(as);
bs.resize(n); fft(bs);
for (int i = 0; i < n; ++i) as[i] *= bs[i];
invFft(as);
as.resize(len);
return as;
}
vector<Mint> square(vector<Mint> as) {
if (as.empty()) return {};
const int len = as.size() + as.size() - 1;
int n = 1;
for (; n < len; n <<= 1) {}
as.resize(n); fft(as);
for (int i = 0; i < n; ++i) as[i] *= as[i];
invFft(as);
as.resize(len);
return as;
}
////////////////////////////////////////////////////////////////////////////////
////////////////////////////////////////////////////////////////////////////////
// inv: log, exp, pow
// fac: shift
// invFac: shift
constexpr int LIM_INV = 1 << 20; // @
Mint inv[LIM_INV], fac[LIM_INV], invFac[LIM_INV];
struct ModIntPreparator {
ModIntPreparator() {
inv[1] = 1;
for (int i = 2; i < LIM_INV; ++i) inv[i] = -((Mint::M / i) * inv[Mint::M % i]);
fac[0] = 1;
for (int i = 1; i < LIM_INV; ++i) fac[i] = fac[i - 1] * i;
invFac[0] = 1;
for (int i = 1; i < LIM_INV; ++i) invFac[i] = invFac[i - 1] * inv[i];
}
} preparator;
// polyWork0: *, inv, div, divAt, log, exp, pow, sqrt, shift
// polyWork1: inv, div, divAt, log, exp, pow, sqrt, shift
// polyWork2: divAt, exp, pow, sqrt
// polyWork3: exp, pow, sqrt
static constexpr int LIM_POLY = 1 << 20; // @
static_assert(LIM_POLY <= 1 << FFT_MAX, "Poly: LIM_POLY <= 1 << FFT_MAX must hold.");
static Mint polyWork0[LIM_POLY], polyWork1[LIM_POLY], polyWork2[LIM_POLY], polyWork3[LIM_POLY];
struct Poly : public vector<Mint> {
Poly() {}
explicit Poly(int n) : vector<Mint>(n) {}
Poly(const vector<Mint> &vec) : vector<Mint>(vec) {}
Poly(std::initializer_list<Mint> il) : vector<Mint>(il) {}
int size() const { return vector<Mint>::size(); }
Mint at(long long k) const { return (0 <= k && k < size()) ? (*this)[k] : 0U; }
int ord() const { for (int i = 0; i < size(); ++i) if ((*this)[i]) return i; return -1; }
int deg() const { for (int i = size(); --i >= 0; ) if ((*this)[i]) return i; return -1; }
Poly mod(int n) const { return Poly(vector<Mint>(data(), data() + min(n, size()))); }
friend std::ostream &operator<<(std::ostream &os, const Poly &fs) {
os << "[";
for (int i = 0; i < fs.size(); ++i) { if (i > 0) os << ", "; os << fs[i]; }
return os << "]";
}
Poly &operator+=(const Poly &fs) {
if (size() < fs.size()) resize(fs.size());
for (int i = 0; i < fs.size(); ++i) (*this)[i] += fs[i];
return *this;
}
Poly &operator-=(const Poly &fs) {
if (size() < fs.size()) resize(fs.size());
for (int i = 0; i < fs.size(); ++i) (*this)[i] -= fs[i];
return *this;
}
// 3 E(|t| + |f|)
Poly &operator*=(const Poly &fs) {
if (empty() || fs.empty()) return *this = {};
const int nt = size(), nf = fs.size();
int n = 1;
for (; n < nt + nf - 1; n <<= 1) {}
assert(n <= LIM_POLY);
resize(n);
fft(data(), n); // 1 E(n)
memcpy(polyWork0, fs.data(), nf * sizeof(Mint));
memset(polyWork0 + nf, 0, (n - nf) * sizeof(Mint));
fft(polyWork0, n); // 1 E(n)
for (int i = 0; i < n; ++i) (*this)[i] *= polyWork0[i];
invFft(data(), n); // 1 E(n)
resize(nt + nf - 1);
return *this;
}
// 13 E(deg(t) - deg(f) + 1)
// rev(t) = rev(f) rev(q) + x^(deg(t)-deg(f)+1) rev(r)
Poly &operator/=(const Poly &fs) {
const int m = deg(), n = fs.deg();
assert(n != -1);
if (m < n) return *this = {};
Poly tsRev(m - n + 1), fsRev(min(m - n, n) + 1);
for (int i = 0; i <= m - n; ++i) tsRev[i] = (*this)[m - i];
for (int i = 0, i0 = min(m - n, n); i <= i0; ++i) fsRev[i] = fs[n - i];
const Poly qsRev = tsRev.div(fsRev, m - n + 1); // 13 E(m - n + 1)
resize(m - n + 1);
for (int i = 0; i <= m - n; ++i) (*this)[i] = qsRev[m - n - i];
return *this;
}
// 13 E(deg(t) - deg(f) + 1) + 3 E(|t|)
Poly &operator%=(const Poly &fs) {
const Poly qs = *this / fs; // 13 E(deg(t) - deg(f) + 1)
*this -= fs * qs; // 3 E(|t|)
resize(deg() + 1);
return *this;
}
Poly &operator*=(const Mint &a) {
for (int i = 0; i < size(); ++i) (*this)[i] *= a;
return *this;
}
Poly &operator/=(const Mint &a) {
const Mint b = a.inv();
for (int i = 0; i < size(); ++i) (*this)[i] *= b;
return *this;
}
Poly operator+() const { return *this; }
Poly operator-() const {
Poly fs(size());
for (int i = 0; i < size(); ++i) fs[i] = -(*this)[i];
return fs;
}
Poly operator+(const Poly &fs) const { return (Poly(*this) += fs); }
Poly operator-(const Poly &fs) const { return (Poly(*this) -= fs); }
Poly operator*(const Poly &fs) const { return (Poly(*this) *= fs); }
Poly operator/(const Poly &fs) const { return (Poly(*this) /= fs); }
Poly operator%(const Poly &fs) const { return (Poly(*this) %= fs); }
Poly operator*(const Mint &a) const { return (Poly(*this) *= a); }
Poly operator/(const Mint &a) const { return (Poly(*this) /= a); }
friend Poly operator*(const Mint &a, const Poly &fs) { return fs * a; }
// 10 E(n)
// f <- f - (t f - 1) f
Poly inv(int n) const {
assert(!empty()); assert((*this)[0]); assert(1 <= n);
assert(n == 1 || 1 << (32 - __builtin_clz(n - 1)) <= LIM_POLY);
Poly fs(n);
fs[0] = (*this)[0].inv();
for (int m = 1; m < n; m <<= 1) {
memcpy(polyWork0, data(), min(m << 1, size()) * sizeof(Mint));
memset(polyWork0 + min(m << 1, size()), 0, ((m << 1) - min(m << 1, size())) * sizeof(Mint));
fft(polyWork0, m << 1); // 2 E(n)
memcpy(polyWork1, fs.data(), min(m << 1, n) * sizeof(Mint));
memset(polyWork1 + min(m << 1, n), 0, ((m << 1) - min(m << 1, n)) * sizeof(Mint));
fft(polyWork1, m << 1); // 2 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m << 1); // 2 E(n)
memset(polyWork0, 0, m * sizeof(Mint));
fft(polyWork0, m << 1); // 2 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m << 1); // 2 E(n)
for (int i = m, i0 = min(m << 1, n); i < i0; ++i) fs[i] = -polyWork0[i];
}
return fs;
}
// 13 E(n)
// g = (1 / f) mod x^m
// h <- h - (f h - t) g
Poly div(const Poly &fs, int n) const {
assert(!fs.empty()); assert(fs[0]); assert(1 <= n);
if (n == 1) return {at(0) / fs[0]};
// m < n <= 2 m
const int m = 1 << (31 - __builtin_clz(n - 1));
assert(m << 1 <= LIM_POLY);
Poly gs = fs.inv(m); // 5 E(n)
gs.resize(m << 1);
fft(gs.data(), m << 1); // 1 E(n)
memcpy(polyWork0, data(), min(m, size()) * sizeof(Mint));
memset(polyWork0 + min(m, size()), 0, ((m << 1) - min(m, size())) * sizeof(Mint));
fft(polyWork0, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= gs[i];
invFft(polyWork0, m << 1); // 1 E(n)
Poly hs(n);
memcpy(hs.data(), polyWork0, m * sizeof(Mint));
memset(polyWork0 + m, 0, m * sizeof(Mint));
fft(polyWork0, m << 1); // 1 E(n)
memcpy(polyWork1, fs.data(), min(m << 1, fs.size()) * sizeof(Mint));
memset(polyWork1 + min(m << 1, fs.size()), 0, ((m << 1) - min(m << 1, fs.size())) * sizeof(Mint));
fft(polyWork1, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= polyWork1[i];
invFft(polyWork0, m << 1); // 1 E(n)
memset(polyWork0, 0, m * sizeof(Mint));
for (int i = m, i0 = min(m << 1, size()); i < i0; ++i) polyWork0[i] -= (*this)[i];
fft(polyWork0, m << 1); // 1 E(n)
for (int i = 0; i < m << 1; ++i) polyWork0[i] *= gs[i];
invFft(polyWork0, m << 1); // 1 E(n)
for (int i = m; i < n; ++i) hs[i] = -polyWork0[i];
return hs;
}
};
struct SubproductTree {
int logN, n, nn;
vector<Mint> xs;
// [DFT_4((X-xs[0])(X-xs[1])(X-xs[2])(X-xs[3]))] [(X-xs[0])(X-xs[1])(X-xs[2])(X-xs[3])mod X^4]
// [ DFT_4((X-xs[0])(X-xs[1])) ] [ DFT_4((X-xs[2])(X-xs[3])) ]
// [ DFT_2(X-xs[0]) ] [ DFT_2(X-xs[1]) ] [ DFT_2(X-xs[2]) ] [ DFT_2(X-xs[3]) ]
vector<Mint> buf;
vector<Mint *> gss;
// (1 - xs[0] X) ... (1 - xs[nn-1] X)
Poly all;
// (ceil(log_2 n) + O(1)) E(n)
SubproductTree(const vector<Mint> &xs_) {
n = xs_.size();
for (logN = 0, nn = 1; nn < n; ++logN, nn <<= 1) {}
xs.assign(nn, 0U);
memcpy(xs.data(), xs_.data(), n * sizeof(Mint));
buf.assign((logN + 1) * (nn << 1), 0U);
gss.assign(nn << 1, nullptr);
for (int h = 0; h <= logN; ++h) for (int u = 1 << h; u < 1 << (h + 1); ++u) {
gss[u] = buf.data() + (h * (nn << 1) + ((u - (1 << h)) << (logN - h + 1)));
}
for (int i = 0; i < nn; ++i) {
gss[nn + i][0] = -xs[i] + 1;
gss[nn + i][1] = -xs[i] - 1;
}
if (nn == 1) gss[1][1] += 2;
for (int h = logN; --h >= 0; ) {
const int m = 1 << (logN - h);
for (int u = 1 << (h + 1); --u >= 1 << h; ) {
for (int i = 0; i < m; ++i) gss[u][i] = gss[u << 1][i] * gss[u << 1 | 1][i];
memcpy(gss[u] + m, gss[u], m * sizeof(Mint));
invFft(gss[u] + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
if (h > 0) {
gss[u][m] -= 2;
const Mint a = FFT_ROOTS[logN - h + 1];
Mint aa = 1;
for (int i = m; i < m << 1; ++i) { gss[u][i] *= aa; aa *= a; };
fft(gss[u] + m, m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
}
}
}
all.resize(nn + 1);
all[0] = 1;
for (int i = 1; i < nn; ++i) all[i] = gss[1][nn + nn - i];
all[nn] = gss[1][nn] - 1;
}
// ((3/2) ceil(log_2 n) + O(1)) E(n) + 10 E(|f|) + 3 E(|f| + 2^(ceil(log_2 n)))
vector<Mint> multiEval(const Poly &fs) const {
vector<Mint> work0(nn), work1(nn), work2(nn);
{
const int m = max(fs.size(), 1);
auto invAll = all.inv(m); // 10 E(|f|)
std::reverse(invAll.begin(), invAll.end());
int mm;
for (mm = 1; mm < m - 1 + nn; mm <<= 1) {}
invAll.resize(mm, 0U);
fft(invAll); // E(|f| + 2^(ceil(log_2 n)))
vector<Mint> ffs(mm, 0U);
memcpy(ffs.data(), fs.data(), fs.size() * sizeof(Mint));
fft(ffs); // E(|f| + 2^(ceil(log_2 n)))
for (int i = 0; i < mm; ++i) ffs[i] *= invAll[i];
invFft(ffs); // E(|f| + 2^(ceil(log_2 n)))
memcpy(((logN & 1) ? work1 : work0).data(), ffs.data() + m - 1, nn * sizeof(Mint));
}
for (int h = 0; h < logN; ++h) {
const int m = 1 << (logN - h);
for (int u = 1 << h; u < 1 << (h + 1); ++u) {
Mint *hs = (((logN - h) & 1) ? work1 : work0).data() + ((u - (1 << h)) << (logN - h));
Mint *hs0 = (((logN - h) & 1) ? work0 : work1).data() + ((u - (1 << h)) << (logN - h));
Mint *hs1 = hs0 + (m >> 1);
fft(hs, m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
for (int i = 0; i < m; ++i) work2[i] = gss[u << 1 | 1][i] * hs[i];
invFft(work2.data(), m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
memcpy(hs0, work2.data() + (m >> 1), (m >> 1) * sizeof(Mint));
for (int i = 0; i < m; ++i) work2[i] = gss[u << 1][i] * hs[i];
invFft(work2.data(), m); // ((1/2) ceil(log_2 n) + O(1)) E(n)
memcpy(hs1, work2.data() + (m >> 1), (m >> 1) * sizeof(Mint));
}
}
work0.resize(n);
return work0;
}
};
////////////////////////////////////////////////////////////////////////////////
constexpr int FACTORIAL_STEP = 10'000'000;
constexpr ModInt<998244353> FACTORIAL[] = {1,295201906,160030060,957629942,545208507,213689172,760025067,939830261,506268060,39806322,808258749,440133909,686156489,741797144,390377694,12629586,544711799,104121967,495867250,421290700,117153405,57084755,202713771,675932866,79781699,956276337,652678397,35212756,655645460,468129309,761699708,533047427,287671032,206068022,50865043,144980423,111276893,259415897,444094191,593907889,573994984,892454686,566073550,128761001,888483202,251718753,548033568,428105027,742756734,546182474,62402409,102052166,826426395,159186619,926316039,176055335,51568171,414163604,604947226,681666415,511621808,924112080,265769800,955559118,763148293,472709375,19536133,860830935,290471030,851685235,242726978,169855231,612759169,599797734,961628039,953297493,62806842,37844313,909741023,689361523,887890124,380694152,669317759,367270918,806951470,843736533,377403437,945260111,786127243,80918046,875880304,364983542,623250998,598764068,804930040,24257676,214821357,791011898,954947696,183092975,};
template <unsigned M> ModInt<M> factorial(long long n) {
assert(n >= 0);
if (n >= static_cast<long long>(M)) return 0;
const long long pos = n / FACTORIAL_STEP;
const long long m0 = pos * FACTORIAL_STEP;
const long long m1 = m0 + FACTORIAL_STEP;
if (m1 < static_cast<long long>(M) && n - m0 > m1 - n) {
ModInt<M> prod = 1;
for (long long i = m1; i > n; ) prod *= i--;
return FACTORIAL[pos + 1] / prod;
} else {
ModInt<M> prod = FACTORIAL[pos];
for (long long i = m0; i < n; ) prod *= ++i;
return prod;
}
}
#line 1 "verify/verify-unit-test/polynomial-matrix-prod.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
//
#line 2 "template/template.hpp"
using namespace std;
// intrinstic
#include <immintrin.h>
#include <algorithm>
#include <array>
#include <bitset>
#include <cassert>
#include <cctype>
#include <cfenv>
#include <cfloat>
#include <chrono>
#include <cinttypes>
#include <climits>
#include <cmath>
#include <complex>
#include <cstdarg>
#include <cstddef>
#include <cstdint>
#include <cstdio>
#include <cstdlib>
#include <cstring>
#include <deque>
#include <fstream>
#include <functional>
#include <initializer_list>
#include <iomanip>
#include <ios>
#include <iostream>
#include <istream>
#include <iterator>
#include <limits>
#include <list>
#include <map>
#include <memory>
#include <new>
#include <numeric>
#include <ostream>
#include <queue>
#include <random>
#include <set>
#include <sstream>
#include <stack>
#include <streambuf>
#include <string>
#include <tuple>
#include <type_traits>
#include <typeinfo>
#include <unordered_map>
#include <unordered_set>
#include <utility>
#include <vector>
namespace nyaan {
// utility
#line 1 "template/util.hpp"
namespace Nyaan {
using ll = long long;
using i64 = long long;
using u64 = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <typename T>
using V = vector<T>;
template <typename T>
using VV = vector<vector<T>>;
using vi = vector<int>;
using vl = vector<long long>;
using vd = V<double>;
using vs = V<string>;
using vvi = vector<vector<int>>;
using vvl = vector<vector<long long>>;
template <typename T, typename U>
struct P : pair<T, U> {
template <typename... Args>
P(Args... args) : pair<T, U>(args...) {}
using pair<T, U>::first;
using pair<T, U>::second;
T &x() { return first; }
const T &x() const { return first; }
U &y() { return second; }
const U &y() const { return second; }
P &operator+=(const P &r) {
first += r.first;
second += r.second;
return *this;
}
P &operator-=(const P &r) {
first -= r.first;
second -= r.second;
return *this;
}
P &operator*=(const P &r) {
first *= r.first;
second *= r.second;
return *this;
}
P operator+(const P &r) const { return P(*this) += r; }
P operator-(const P &r) const { return P(*this) -= r; }
P operator*(const P &r) const { return P(*this) *= r; }
};
using pl = P<ll, ll>;
using pi = P<int, int>;
using vp = V<pl>;
constexpr int inf = 1001001001;
constexpr long long infLL = 4004004004004004004LL;
template <typename T>
int sz(const T &t) {
return t.size();
}
template <typename T, typename U>
inline bool amin(T &x, U y) {
return (y < x) ? (x = y, true) : false;
}
template <typename T, typename U>
inline bool amax(T &x, U y) {
return (x < y) ? (x = y, true) : false;
}
template <typename T>
inline T Max(const vector<T> &v) {
return *max_element(begin(v), end(v));
}
template <typename T>
inline T Min(const vector<T> &v) {
return *min_element(begin(v), end(v));
}
template <typename T>
inline long long Sum(const vector<T> &v) {
return accumulate(begin(v), end(v), 0LL);
}
template <typename T>
int lb(const vector<T> &v, const T &a) {
return lower_bound(begin(v), end(v), a) - begin(v);
}
template <typename T>
int ub(const vector<T> &v, const T &a) {
return upper_bound(begin(v), end(v), a) - begin(v);
}
constexpr long long TEN(int n) {
long long ret = 1, x = 10;
for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1);
return ret;
}
template <typename T, typename U>
pair<T, U> mkp(const T &t, const U &u) {
return make_pair(t, u);
}
template <typename T>
vector<T> mkrui(const vector<T> &v, bool rev = false) {
vector<T> ret(v.size() + 1);
if (rev) {
for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1];
} else {
for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i];
}
return ret;
};
template <typename T>
vector<T> mkuni(const vector<T> &v) {
vector<T> ret(v);
sort(ret.begin(), ret.end());
ret.erase(unique(ret.begin(), ret.end()), ret.end());
return ret;
}
template <typename F>
vector<int> mkord(int N, F f) {
vector<int> ord(N);
iota(begin(ord), end(ord), 0);
sort(begin(ord), end(ord), f);
return ord;
}
template <typename T>
vector<int> mkinv(vector<T> &v) {
int max_val = *max_element(begin(v), end(v));
vector<int> inv(max_val + 1, -1);
for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i;
return inv;
}
} // namespace Nyaan
#line 58 "template/template.hpp"
// bit operation
#line 1 "template/bitop.hpp"
namespace Nyaan {
__attribute__((target("popcnt"))) inline int popcnt(const u64 &a) {
return _mm_popcnt_u64(a);
}
inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; }
inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; }
template <typename T>
inline int gbit(const T &a, int i) {
return (a >> i) & 1;
}
template <typename T>
inline void sbit(T &a, int i, bool b) {
if (gbit(a, i) != b) a ^= T(1) << i;
}
constexpr long long PW(int n) { return 1LL << n; }
constexpr long long MSK(int n) { return (1LL << n) - 1; }
} // namespace Nyaan
#line 61 "template/template.hpp"
// inout
#line 1 "template/inout.hpp"
namespace Nyaan {
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...);
}
void outr() {}
template <typename T, class... U, char sep = ' '>
void outr(const T &t, const U &... u) {
cout << t;
outr(u...);
}
struct IoSetupNya {
IoSetupNya() {
cin.tie(nullptr);
ios::sync_with_stdio(false);
cout << fixed << setprecision(15);
cerr << fixed << setprecision(7);
}
} iosetupnya;
} // namespace Nyaan
#line 64 "template/template.hpp"
// debug
#line 1 "template/debug.hpp"
namespace DebugImpl {
template <typename U, typename = void>
struct is_specialize : false_type {};
template <typename U>
struct is_specialize<
U, typename conditional<false, typename U::iterator, void>::type>
: true_type {};
template <typename U>
struct is_specialize<
U, typename conditional<false, decltype(U::first), void>::type>
: true_type {};
template <typename U>
struct is_specialize<U, enable_if_t<is_integral<U>::value, void>> : true_type {
};
void dump(const char& t) { cerr << t; }
void dump(const string& t) { cerr << t; }
void dump(const bool& t) { cerr << (t ? "true" : "false"); }
template <typename U,
enable_if_t<!is_specialize<U>::value, nullptr_t> = nullptr>
void dump(const U& t) {
cerr << t;
}
template <typename T>
void dump(const T& t, enable_if_t<is_integral<T>::value>* = nullptr) {
string res;
if (t == Nyaan::inf) res = "inf";
if constexpr (is_signed<T>::value) {
if (t == -Nyaan::inf) res = "-inf";
}
if constexpr (sizeof(T) == 8) {
if (t == Nyaan::infLL) res = "inf";
if constexpr (is_signed<T>::value) {
if (t == -Nyaan::infLL) res = "-inf";
}
}
if (res.empty()) res = to_string(t);
cerr << res;
}
template <typename T, typename U>
void dump(const pair<T, U>&);
template <typename T>
void dump(const pair<T*, int>&);
template <typename T>
void dump(const T& t,
enable_if_t<!is_void<typename T::iterator>::value>* = nullptr) {
cerr << "[ ";
for (auto it = t.begin(); it != t.end();) {
dump(*it);
cerr << (++it == t.end() ? "" : ", ");
}
cerr << " ]";
}
template <typename T, typename U>
void dump(const pair<T, U>& t) {
cerr << "( ";
dump(t.first);
cerr << ", ";
dump(t.second);
cerr << " )";
}
template <typename T>
void dump(const pair<T*, int>& t) {
cerr << "[ ";
for (int i = 0; i < t.second; i++) {
dump(t.first[i]);
cerr << (i == t.second - 1 ? "" : ", ");
}
cerr << " ]";
}
void trace() { cerr << endl; }
template <typename Head, typename... Tail>
void trace(Head&& head, Tail&&... tail) {
cerr << " ";
dump(head);
if (sizeof...(tail) != 0) cerr << ",";
trace(forward<Tail>(tail)...);
}
} // namespace DebugImpl
#ifdef NyaanDebug
#define trc(...) \
do { \
cerr << "## " << #__VA_ARGS__ << " = "; \
DebugImpl::trace(__VA_ARGS__); \
} while (0)
#else
#define trc(...) (void(0))
#endif
#line 67 "template/template.hpp"
// macro
#line 1 "template/macro.hpp"
#define each(x, v) for (auto&& x : v)
#define each2(x, y, v) for (auto&& [x, y] : v)
#define all(v) (v).begin(), (v).end()
#define rep(i, N) for (long long i = 0; i < (long long)(N); i++)
#define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--)
#define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++)
#define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--)
#define reg(i, a, b) for (long long i = (a); i < (b); i++)
#define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--)
#define fi first
#define se second
#define ini(...) \
int __VA_ARGS__; \
in(__VA_ARGS__)
#define inl(...) \
long long __VA_ARGS__; \
in(__VA_ARGS__)
#define ins(...) \
string __VA_ARGS__; \
in(__VA_ARGS__)
#define in2(s, t) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i]); \
}
#define in3(s, t, u) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i]); \
}
#define in4(s, t, u, v) \
for (int i = 0; i < (int)s.size(); i++) { \
in(s[i], t[i], u[i], v[i]); \
}
#define die(...) \
do { \
Nyaan::out(__VA_ARGS__); \
return; \
} while (0)
#line 70 "template/template.hpp"
namespace Nyaan {
void solve();
}
int main() { Nyaan::solve(); }
#line 4 "verify/verify-unit-test/polynomial-matrix-prod.test.cpp"
//
#line 2 "fps/ntt-friendly-fps.hpp"
#line 2 "ntt/ntt.hpp"
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), t(M);
for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i];
for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i];
fft4(s, k);
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"
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;
}
FPS pre(int sz) const {
return FPS(begin(*this), begin(*this) + min((int)this->size(), sz));
}
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)[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;
/**
* @brief 多項式/形式的冪級数ライブラリ
* @docs docs/fps/formal-power-series.md
*/
#line 5 "fps/ntt-friendly-fps.hpp"
template <typename mint>
void FormalPowerSeries<mint>::set_fft() {
if (!ntt_ptr) ntt_ptr = new NTT<mint>;
}
template <typename mint>
FormalPowerSeries<mint>& FormalPowerSeries<mint>::operator*=(
const FormalPowerSeries<mint>& r) {
if (this->empty() || r.empty()) {
this->clear();
return *this;
}
set_fft();
auto ret = static_cast<NTT<mint>*>(ntt_ptr)->multiply(*this, r);
return *this = FormalPowerSeries<mint>(ret.begin(), ret.end());
}
template <typename mint>
void FormalPowerSeries<mint>::ntt() {
set_fft();
static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}
template <typename mint>
void FormalPowerSeries<mint>::intt() {
set_fft();
static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}
template <typename mint>
void FormalPowerSeries<mint>::ntt_doubling() {
set_fft();
static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}
template <typename mint>
int FormalPowerSeries<mint>::ntt_pr() {
set_fft();
return static_cast<NTT<mint>*>(ntt_ptr)->pr;
}
template <typename mint>
FormalPowerSeries<mint> FormalPowerSeries<mint>::inv(int deg) const {
assert((*this)[0] != mint(0));
if (deg == -1) deg = (int)this->size();
FormalPowerSeries<mint> res(deg);
res[0] = {mint(1) / (*this)[0]};
for (int d = 1; d < deg; d <<= 1) {
FormalPowerSeries<mint> f(2 * d), g(2 * d);
for (int j = 0; j < min((int)this->size(), 2 * d); j++) f[j] = (*this)[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> FormalPowerSeries<mint>::exp(int deg) const {
using fps = FormalPowerSeries<mint>;
assert((*this).size() == 0 || (*this)[0] == mint(0));
if (deg == -1) deg = this->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)this->size() ? (*this)[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(*this), begin(*this) + min<int>(this->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>(this->size(), 2 * m); ++i) x[i] += (*this)[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
* @docs docs/fps/ntt-friendly-fps.md
*/
#line 2 "matrix/polynomial-matrix-prefix-prod.hpp"
#line 2 "fps/sample-point-shift.hpp"
#line 2 "modulo/binomial.hpp"
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});
while (MAX >= (int)f.size()) extend();
}
void extend() {
int n = f.size();
int m = n * 2;
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);
}
T H(int n, int r) {
if (n < 0 || r < 0) return T(0);
return r == 0 ? 1 : C(n + r - 1, r);
}
};
#line 5 "fps/sample-point-shift.hpp"
// input : y(0), y(1), ..., y(n - 1)
// output : y(t), y(t + 1), ..., y(t + m - 1)
// (if m is default, m = n)
template <typename mint>
FormalPowerSeries<mint> SamplePointShift(FormalPowerSeries<mint>& y, mint t,
int m = -1) {
if (m == -1) m = y.size();
long long T = t.get();
int k = (int)y.size() - 1;
T %= mint::get_mod();
if (T <= k) {
FormalPowerSeries<mint> ret(m);
int ptr = 0;
for (int64_t i = T; i <= k and ptr < m; i++) {
ret[ptr++] = y[i];
}
if (k + 1 < T + m) {
auto suf = SamplePointShift<mint>(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::get_mod()) {
auto pref = SamplePointShift<mint>(y, T, mint::get_mod() - T);
auto suf = SamplePointShift<mint>(y, 0, m - pref.size());
copy(begin(suf), end(suf), back_inserter(pref));
return pref;
}
FormalPowerSeries<mint> finv(k + 1, 1), d(k + 1);
for (int i = 2; i <= k; i++) finv[k] *= i;
finv[k] = mint(1) / finv[k];
for (int i = k; i >= 1; i--) finv[i - 1] = finv[i] * i;
for (int i = 0; i <= k; i++) {
d[i] = finv[i] * finv[k - i] * y[i];
if ((k - i) & 1) d[i] = -d[i];
}
FormalPowerSeries<mint> h(m + k);
for (int i = 0; i < m + k; i++) {
h[i] = mint(1) / (T - k + i);
}
auto dh = d * h;
FormalPowerSeries<mint> 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;
}
#line 2 "matrix/matrix.hpp"
template <class T>
struct Matrix {
vector<vector<T> > A;
Matrix() = default;
Matrix(int n, int m) : A(n, vector<T>(m, T())) {}
Matrix(int n) : A(n, vector<T>(n, T())){};
int H() const { return A.size(); }
int W() const { return A[0].size(); }
int size() const { return A.size(); }
inline const vector<T> &operator[](int k) const { return A[k]; }
inline vector<T> &operator[](int k) { return A[k]; }
static Matrix I(int n) {
Matrix mat(n);
for (int i = 0; i < n; i++) mat[i][i] = 1;
return (mat);
}
Matrix &operator+=(const Matrix &B) {
int n = H(), m = W();
assert(n == B.H() && m == B.W());
for (int i = 0; i < n; i++)
for (int j = 0; j < m; j++) (*this)[i][j] += B[i][j];
return (*this);
}
Matrix &operator-=(const Matrix &B) {
int n = H(), m = W();
assert(n == B.H() && m == B.W());
for (int i = 0; i < n; i++)
for (int j = 0; j < m; j++) (*this)[i][j] -= B[i][j];
return (*this);
}
Matrix &operator*=(const Matrix &B) {
int n = H(), m = B.W(), p = W();
assert(p == B.H());
vector<vector<T> > C(n, vector<T>(m, T{}));
for (int i = 0; i < n; i++)
for (int k = 0; k < p; k++)
for (int j = 0; j < m; j++) C[i][j] += (*this)[i][k] * B[k][j];
A.swap(C);
return (*this);
}
Matrix &operator^=(long long k) {
Matrix B = Matrix::I(H());
while (k > 0) {
if (k & 1) B *= *this;
*this *= *this;
k >>= 1LL;
}
A.swap(B.A);
return (*this);
}
Matrix operator+(const Matrix &B) const { return (Matrix(*this) += B); }
Matrix operator-(const Matrix &B) const { return (Matrix(*this) -= B); }
Matrix operator*(const Matrix &B) const { return (Matrix(*this) *= B); }
Matrix operator^(const long long k) const { return (Matrix(*this) ^= k); }
bool operator==(const Matrix &B) const {
assert(H() == B.H() && W() == B.W());
for (int i = 0; i < H(); i++)
for (int j = 0; j < W(); j++)
if (A[i][j] != B[i][j]) return false;
return true;
}
bool operator!=(const Matrix &B) const {
assert(H() == B.H() && W() == B.W());
for (int i = 0; i < H(); i++)
for (int j = 0; j < W(); j++)
if (A[i][j] != B[i][j]) return true;
return false;
}
friend ostream &operator<<(ostream &os, const Matrix &p) {
int n = p.H(), m = p.W();
for (int i = 0; i < n; i++) {
os << (i ? " " : "") << "[";
for (int j = 0; j < m; j++) {
os << p[i][j] << (j + 1 == m ? "]\n" : ",");
}
}
return (os);
}
T determinant() const {
Matrix B(*this);
assert(H() == W());
T ret = 1;
for (int i = 0; i < H(); i++) {
int idx = -1;
for (int j = i; j < W(); j++) {
if (B[j][i] != 0) {
idx = j;
break;
}
}
if (idx == -1) return 0;
if (i != idx) {
ret *= T(-1);
swap(B[i], B[idx]);
}
ret *= B[i][i];
T inv = T(1) / B[i][i];
for (int j = 0; j < W(); j++) {
B[i][j] *= inv;
}
for (int j = i + 1; j < H(); j++) {
T a = B[j][i];
if (a == 0) continue;
for (int k = i; k < W(); k++) {
B[j][k] -= B[i][k] * a;
}
}
}
return ret;
}
};
/**
* @brief 行列ライブラリ
*/
#line 6 "matrix/polynomial-matrix-prefix-prod.hpp"
// return m(k-1) * m(k-2) * ... * m(1) * m(0)
template <typename mint>
Matrix<mint> polynomial_matrix_prod(Matrix<FormalPowerSeries<mint>> &m,
long long k) {
using Mat = Matrix<mint>;
using fps = FormalPowerSeries<mint>;
auto shift = [](vector<Mat> &G, mint x) -> vector<Mat> {
int d = G.size(), n = G[0].size();
vector<Mat> H(d, Mat(n));
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
fps g(d);
for (int l = 0; l < d; l++) g[l] = G[l][i][j];
fps h = SamplePointShift(g, x);
for (int l = 0; l < d; l++) H[l][i][j] = h[l];
}
}
return H;
};
int n = m.size();
int deg = 1;
for (auto &_ : m.A) {
for (auto &x : _) deg = max<int>(deg, (int)x.size() - 1);
}
while (deg & (deg - 1)) deg++;
vector<Mat> G(deg + 1);
long long v = 1;
while (deg * v * v < k) v *= 2;
mint iv = mint(v).inverse();
for (int i = 0; i < (int)G.size(); i++) {
mint x = mint(v) * i;
Mat mt(n);
for (int j = 0; j < n; j++)
for (int l = 0; l < n; l++) mt[j][l] = m[j][l].eval(x);
G[i] = mt;
}
for (long long w = 1; w != v; w <<= 1) {
mint W = w;
auto G1 = shift(G, W * iv);
auto G2 = shift(G, (W * deg * v + v) * iv);
auto G3 = shift(G, (W * deg * v + v + W) * iv);
for (int i = 0; i <= w * deg; i++)
G[i] = G1[i] * G[i], G2[i] = G3[i] * G2[i];
copy(begin(G2), end(G2) - 1, back_inserter(G));
}
Mat res = Mat::I(n);
long long i = 0;
while (i + v <= k) res = G[i / v] * res, i += v;
while (i < k) {
Mat mt(n);
for (int j = 0; j < n; j++)
for (int l = 0; l < n; l++) mt[j][l] = m[j][l].eval(i);
res = mt * res;
i++;
}
return res;
}
/**
* @brief 多項式行列のprefix product
*/
#line 2 "misc/rng.hpp"
namespace my_rand {
using i64 = long long;
using u64 = unsigned long long;
// [0, 2^64 - 1)
u64 rng() {
static u64 _x =
u64(chrono::duration_cast<chrono::nanoseconds>(
chrono::high_resolution_clock::now().time_since_epoch())
.count()) *
10150724397891781847ULL;
_x ^= _x << 7;
return _x ^= _x >> 9;
}
// [l, r]
i64 rng(i64 l, i64 r) {
assert(l <= r);
return l + rng() % (r - l + 1);
}
// [l, r)
i64 randint(i64 l, i64 r) {
assert(l < r);
return l + rng() % (r - l);
}
// choose n numbers from [l, r) without overlapping
vector<i64> randset(i64 l, i64 r, i64 n) {
assert(l <= r && n <= r - l);
unordered_set<i64> s;
for (i64 i = n; i; --i) {
i64 m = randint(l, r + 1 - i);
if (s.find(m) != s.end()) m = r - i;
s.insert(m);
}
vector<i64> ret;
for (auto& x : s) ret.push_back(x);
return ret;
}
// [0.0, 1.0)
double rnd() { return rng() * 5.42101086242752217004e-20; }
template <typename T>
void randshf(vector<T>& v) {
int n = v.size();
for (int i = 1; i < n; i++) swap(v[i], v[randint(0, i + 1)]);
}
} // namespace my_rand
using my_rand::randint;
using my_rand::randset;
using my_rand::randshf;
using my_rand::rnd;
using my_rand::rng;
#line 2 "modint/montgomery-modint.hpp"
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(r * mod == 1, "invalid, r * mod != 1");
static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30");
static_assert((mod & 1) == 1, "invalid, mod % 2 == 0");
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 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 { return pow(mod - 2); }
friend ostream &operator<<(ostream &os, const mint &b) {
return os << b.get();
}
friend istream &operator>>(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 9 "verify/verify-unit-test/polynomial-matrix-prod.test.cpp"
//
using namespace Nyaan;
using mint = LazyMontgomeryModInt<998244353>;
using fps = FormalPowerSeries<mint>;
using fmat = Matrix<fps>;
using mat = Matrix<mint>;
mat eval(fmat& f, mint x) {
mat m(f.H(), f.W());
rep(i, f.H()) rep(j, f.W()) m[i][j] = f[i][j].eval(x);
return m;
}
void test() {
int n = randint(1, 6);
int d = randint(1, 11);
// cerr << " n : " << n << " d : " << d << endl;
fmat m(n);
rep(i, n) rep(j, n) {
fps f(d);
each(x, f) x = rng();
m[i][j] = f;
}
mat prod = mat::I(n);
rep(k, 1000) {
// if(k % 200 == 0 and k) cerr << k << " finished." << endl;
mat m2 = polynomial_matrix_prod(m, k);
assert(prod == m2);
prod = eval(m, k) * prod;
}
// cerr << "ok" << endl;
}
void Nyaan::solve() {
test();
int a, b;
cin >> a >> b;
cout << a + b << endl;
}
} // nyaan
Mint brute(Int n, Int k) {
Mint ret = 0;
Mint pw = 1;
for (int a2 = 0; a2 <= k / 2; ++a2) {
if (a2 >= k - n) {
Mint num = 1;
num *= fac[n] * invFac[n - (k - a2)] * invFac[k - 2 * a2] * invFac[a2];
num *= pw;
ret += num;
}
pw *= inv[8];
}
ret *= fac[k];
ret *= Mint(2).pow(k);
return ret;
}
Mint slow(Int n, Int k) {
if (k >= MO) {
return 0;
}
const Int a0 = max(k - n, 0LL);
const Int a1 = k / 2;
assert(a0 <= a1);
// n (n - 1) ... (n - (k - a1) + 1)
Mint f = 1;
for (int i = 0; i < k - a1; ++i) {
f *= (n - i);
}
f *= invFac[a1];
f *= inv[8].pow(a1);
Mint ret = f;
for (Int a = a1; a > a0; --a) {
// a -> a-1
f = f * 8 * (n - k + a) * a * inv[k - 2 * a + 1] * inv[k - 2 * a + 2];
ret += f;
}
f = ret;
f *= fac[k];
f *= Mint(2).pow(k);
return f;
}
int T;
vector<Int> N, K;
vector<Mint> ans;
Int maxK;
namespace small {
vector<Mint> bs;
pair<Poly, Poly> solve(int l, int r) {
if (r - l == 1) {
return make_pair(Poly{-l, 1}, Poly{bs[l]});
} else {
const int mid = (l + r) / 2;
const auto resL = solve(l, mid);
const auto resR = solve(mid, r);
return make_pair(resL.first * resR.first, resL.second + resL.first * resR.second);
}
}
void run() {
vector<vector<int>> tss(maxK + 1);
for (int t = 0; t < T; ++t) {
tss[K[t]].push_back(t);
}
// const Int big=1;
const Int big = sqrt(maxK) * log1p(maxK);
for (int k = 0; k <= maxK; ++k) {
const auto &ts = tss[k];
const int tsLen = ts.size();
if (tsLen >= big) {
bs.assign(k + 1, 0);
{
Mint pw = fac[k] * Mint(2).pow(k);
for (int a2 = 0; a2 <= k / 2; ++a2) {
bs[k - a2] += invFac[k - 2 * a2] * invFac[a2] * pw;
pw *= inv[8];
}
}
const auto res = solve(0, k + 1);
vector<Mint> ns(tsLen);
for (int i = 0; i < tsLen; ++i) {
ns[i] = N[ts[i]];
}
const auto ys = SubproductTree(ns).multiEval(res.second);
for (int i = 0; i < tsLen; ++i) {
ans[ts[i]] = ys[i];
}
} else {
for (const int t : ts) {
ans[t] = slow(N[t], k);
}
}
}
}
} // small
namespace large {
Mint solve(Int n, Int k) {
if (k >= MO) {
return 0;
}
const Int a0 = max(k - n, 0LL);
const Int a1 = k / 2;
assert(a0 <= a1);
// n (n - 1) ... (n - (k - a1) + 1)
if (n / MO != (n - (k - a1)) / MO) {
return 0;
}
Mint f = factorial<MO>(n % MO) / factorial<MO>((n - (k - a1)) % MO);
f *= factorial<MO>(a1).inv();
f *= inv[8].pow(a1);
/*
ret = f;
for (Int a = a1; a > a0; --a) {
// a -> a-1
f = f * 8 * (n - k + a) * a / (k - 2 * a + 1) / (k - 2 * a + 2);
ret += f;
}
*/
{
using namespace nyaan;
using fps = FormalPowerSeries<mint>;
Matrix<fps> m(2);
m[0][0] = m[1][0] = fps{8} * fps{n - k + a1, -1} * fps{a1, -1};
m[1][1] = fps{k + 1 - 2 * a1, 2} * fps{k + 2 - 2 * a1, 2};
// cerr<<n<<" "<<k<<": m = "<<m<<endl;
const auto prod = polynomial_matrix_prod(m, a1 - a0);
f *= Mint((prod[1][0] + prod[1][1]).get());
}
// /= (K - 2 a1 + 1) ... (K - 2 a0)
f *= factorial<MO>(k - 2 * a1) / factorial<MO>(k - 2 * a0);
f *= factorial<MO>(k);
f *= Mint(2).pow(k);
return f;
}
void run() {
for (int t = 0; t < T; ++t) {
ans[t] = solve(N[t], K[t]);
}
}
} // large
int main() {
for (; ~scanf("%d", &T); ) {
N.resize(T);
K.resize(T);
for (int t = 0; t < T; ++t) {
scanf("%lld%lld", &N[t], &K[t]);
}
ans.assign(T, 0);
maxK = *max_element(K.begin(), K.end());
// if(false){
if (maxK <= 100'000) {
small::run();
} else {
large::run();
}
for (int t = 0; t < T; ++t) {
printf("%u\n", ans[t].x);
}
}
return 0;
}