Algorithm-Library
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test/yosupo/Math/Kth-term-of-Linearly-Recurrent-Sequence.test.cpp
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- Last update: 2023-11-11 17:58:53+09:00
- Problem: https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence
Depends on
- math/convolution/ntt.hpp
- math/fps/fps-template.hpp
- math/fps/fps.hpp
- math/fps/kitamasa.hpp
- math/mint.hpp
- template/template.hpp
Code
#define PROBLEM "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"
#include "../../../template/template.hpp"
#include "../../../math/fps/kitamasa.hpp"
using mint = Mint<998244353>;
using fps = FPS<mint>;
int main(){
cin.tie(nullptr);
ios::sync_with_stdio(false);
int d; ll K;
cin >> d >> K;
fps a(d), c(d+1);
rep(i, d) cin >> a[i];
rep(i, d) cin >> c[i+1];
c = fps{1} - c;
cout << kitamasa(K, c, a) << "\n";
}#line 1 "test/yosupo/Math/Kth-term-of-Linearly-Recurrent-Sequence.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/kth_term_of_linearly_recurrent_sequence"
#line 1 "template/template.hpp"
#include <iostream>
#include <cmath>
#include <string>
#include <vector>
#include <algorithm>
#include <tuple>
#include <cstdint>
#include <cstdio>
#include <map>
#include <queue>
#include <set>
#include <stack>
#include <deque>
#include <bitset>
#include <cctype>
#include <climits>
#include <functional>
#include <cassert>
#include <numeric>
#include <cstring>
#define rep(i, n) for(int i = 0; i < (n); i++)
#define per(i, n) for(int i = (n) - 1; i >= 0; i--)
using ll = long long;
#define vi vector<int>
#define vvi vector<vi>
#define vl vector<ll>
#define pii pair<int, int>
#define pll pair<ll, ll>
#define all(a) (a).begin(), (a).end()
#define rall(a) (a).rbegin(), (a).rend()
constexpr int mod = 1000000007;
using namespace std;
template<class T, class U>
bool chmax(T &a, const U &b){ return a < b ? (a = b, 1) : 0; }
template<class T, class U>
bool chmin(T &a, const U &b){ return a > b ? (a = b, 1) : 0; }
#line 4 "test/yosupo/Math/Kth-term-of-Linearly-Recurrent-Sequence.test.cpp"
#line 1 "math/convolution/ntt.hpp"
template <class mint>
struct NTT {
static constexpr uint32_t get_pr(){
const 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(true){
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(const int k){
mint w[level], y[level];
w[k - 1] = mint(pr).pow((mod - 1) / (1 << k));
y[k - 1] = w[k - 1].inv();
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, const 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));
const 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++){
const mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
const mint t0p2 = t0 + t2, t1p3 = t1 + t3;
const 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++){
const mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww, t3 = a[j2 + v] * wx;
const mint t0p2 = t0 + t2, t1p3 = t1 + t3;
const 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, const 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;
const 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++){
const mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3];
const mint t0p1 = t0 + t1, t2p3 = t2 + t3;
const 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++){
const mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v];
const mint t0p1 = t0 + t1, t2p3 = t2 + t3;
const 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()));
const mint iv = mint(a.size()).inv();
for(auto &x : a) x *= iv;
}
vector<mint> multiply(const vector<mint> &a, const vector<mint> &b){
const 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);
const mint invm = mint(M).inv();
for(int i = 0; i < l; i++) s[i] *= invm;
return s;
}
void ntt_doubling(vector<mint> &a){
const 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 "math/mint.hpp"
template <int mod>
struct Mint {
ll x;
constexpr Mint(ll x = 0) : x((x + mod) % mod){}
static constexpr int get_mod(){ return mod; }
constexpr Mint operator-() const{ return Mint(-x); }
constexpr Mint operator+=(const Mint &a){
if((x += a.x) >= mod) x -= mod;
return *this;
}
constexpr Mint &operator++(){
if(++x == mod) x = 0;
return *this;
}
constexpr Mint operator++(int){
Mint temp = *this;
if(++x == mod) x = 0;
return temp;
}
constexpr Mint &operator-=(const Mint &a){
if((x -= a.x) < 0) x += mod;
return *this;
}
constexpr Mint &operator--(){
if(--x < 0) x += mod;
return *this;
}
constexpr Mint operator--(int){
Mint temp = *this;
if(--x < 0) x += mod;
return temp;
}
constexpr Mint &operator*=(const Mint &a){
(x *= a.x) %= mod;
return *this;
}
constexpr Mint operator+(const Mint &a) const{ return Mint(*this) += a; }
constexpr Mint operator-(const Mint &a) const{ return Mint(*this) -= a; }
constexpr Mint operator*(const Mint &a) const{ return Mint(*this) *= a; }
constexpr Mint pow(ll t) const{
if(!t) return 1;
Mint res = 1, v = *this;
while(t){
if(t & 1) res *= v;
v *= v;
t >>= 1;
}
return res;
}
constexpr Mint inv() const{ return pow(mod - 2); }
constexpr Mint &operator/=(const Mint &a){ return (*this) *= a.inv(); }
constexpr Mint operator/(const Mint &a) const{ return Mint(*this) /= a; }
constexpr bool operator==(const Mint &a) const{ return x == a.x; }
constexpr bool operator!=(const Mint &a) const{ return x != a.x; }
constexpr bool operator<(const Mint &a) const{ return x < a.x; }
constexpr bool operator>(const Mint &a) const{ return x > a.x; }
friend istream &operator>>(istream &is, Mint &a){ return is >> a.x; }
friend ostream &operator<<(ostream &os, const Mint &a){ return os << a.x; }
};
//using mint = Mint<1000000007>;
#line 3 "math/fps/fps-template.hpp"
template <class mint>
struct FPS : vector<mint> {
using vector<mint>::vector;
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;
}
const int n = this->size() - r.size() + 1;
if((int)r.size() <= 64){
FPS f(*this), g(r);
g.shrink();
const mint coeff = g.back().inv();
for(auto &x : g) x *= coeff;
const int deg = (int)f.size() - (int)g.size() + 1;
const 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)){
const 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 <class mint>
void *FPS<mint>::ntt_ptr = nullptr;
#line 2 "math/fps/fps.hpp"
template <class mint>
void FPS<mint>::set_fft(){ if(!ntt_ptr) ntt_ptr = new NTT<mint>; }
template <class mint>
FPS<mint> &FPS<mint>::operator*=(const FPS<mint> &r){
if(this->empty() || r.empty()){
this->clear();
return *this;
}
set_fft();
const auto ret = static_cast<NTT<mint>*>(ntt_ptr)->multiply(*this, r);
return *this = FPS<mint>(ret.begin(), ret.end());
}
template <class mint>
void FPS<mint>::ntt(){
set_fft();
static_cast<NTT<mint>*>(ntt_ptr)->ntt(*this);
}
template <class mint>
void FPS<mint>::intt(){
set_fft();
static_cast<NTT<mint>*>(ntt_ptr)->intt(*this);
}
template <class mint>
void FPS<mint>::ntt_doubling(){
set_fft();
static_cast<NTT<mint>*>(ntt_ptr)->ntt_doubling(*this);
}
template <class mint>
int FPS<mint>::ntt_pr(){
set_fft();
return static_cast<NTT<mint>*>(ntt_ptr)->pr;
}
template <class mint>
FPS<mint> FPS<mint>::inv(int deg) const{
assert((*this)[0] != mint(0));
if(deg == -1) deg = (int)this->size();
FPS<mint> res(deg);
res[0] = { mint(1) / (*this)[0] };
for(int d = 1; d < deg; d <<= 1){
FPS<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 <class mint>
FPS<mint> FPS<mint>::exp(int deg) const{
using fps = FPS<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();
const auto mod = mint::get_mod();
while((int)inv.size() <= n){
const 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;
const mint 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 };
}
#line 2 "math/fps/kitamasa.hpp"
// #include "fps-arbitrary-mod.hpp"
template <class mint>
mint LinearRecurrence(ll k, FPS<mint> Q, FPS<mint> P){
Q.shrink();
mint ret = 0;
if(P.size() >= Q.size()){
const auto R = P / Q;
P -= R * Q;
P.shrink();
if(k < (int)R.size()) ret += R[k];
}
if((int)P.size() == 0) return ret;
FPS<mint>::set_fft();
if(FPS<mint>::ntt_ptr == nullptr){
P.resize((int)Q.size() - 1);
while(k){
auto Q2 = Q;
for(int i = 1; i < (int)Q2.size(); i += 2) Q2[i] = -Q2[i];
const auto S = P * Q2;
const auto T = Q * Q2;
if(k & 1){
for(int i = 1; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
for(int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
}else{
for(int i = 0; i < (int)S.size(); i += 2) P[i >> 1] = S[i];
for(int i = 0; i < (int)T.size(); i += 2) Q[i >> 1] = T[i];
}
k >>= 1;
}
return ret + P[0];
}else{
int N = 1;
while(N < (int)Q.size()) N <<= 1;
P.resize(2 * N);
Q.resize(2 * N);
P.ntt();
Q.ntt();
vector<mint> S(2 * N), T(2 * N);
vector<int> btr(N);
for(int i = 0, logn = __builtin_ctz(N); i < (1 << logn); i++){
btr[i] = (btr[i >> 1] >> 1) + ((i & 1) << (logn - 1));
}
const mint dw = mint(FPS<mint>::ntt_pr()).inv().pow((mint::get_mod() - 1) / (2 * N));
while(k){
mint inv2 = mint(2).inv();
// even degree of Q(x)Q(-x)
T.resize(N);
for(int i = 0; i < N; i++) T[i] = Q[(i << 1) | 0] * Q[(i << 1) | 1];
S.resize(N);
if(k & 1){
// odd degree of P(x)Q(-x)
for(auto &i : btr) {
S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] -
P[(i << 1) | 1] * Q[(i << 1) | 0]) * inv2;
inv2 *= dw;
}
}else{
// even degree of P(x)Q(-x)
for(int i = 0; i < N; i++){
S[i] = (P[(i << 1) | 0] * Q[(i << 1) | 1] +
P[(i << 1) | 1] * Q[(i << 1) | 0]) * inv2;
}
}
swap(P, S);
swap(Q, T);
k >>= 1;
if(k < N) break;
P.ntt_doubling();
Q.ntt_doubling();
}
P.intt();
Q.intt();
return ret + (P * (Q.inv()))[k];
}
}
template <class mint>
mint kitamasa(ll N, FPS<mint> Q, FPS<mint> a){
assert(!Q.empty() && Q[0] != 0);
if(N < (int)a.size()) return a[N];
assert((int)a.size() >= int(Q.size()) - 1);
auto P = a.pre((int)Q.size() - 1) * Q;
P.resize(Q.size() - 1);
return LinearRecurrence<mint>(N, Q, P);
}
#line 6 "test/yosupo/Math/Kth-term-of-Linearly-Recurrent-Sequence.test.cpp"
using mint = Mint<998244353>;
using fps = FPS<mint>;
int main(){
cin.tie(nullptr);
ios::sync_with_stdio(false);
int d; ll K;
cin >> d >> K;
fps a(d), c(d+1);
rep(i, d) cin >> a[i];
rep(i, d) cin >> c[i+1];
c = fps{1} - c;
cout << kitamasa(K, c, a) << "\n";
}