#line 1 "verify/fps/LC_sum_of_exponential_times_polynomial_limit.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial_limit"
#line 2 "template/template.hpp"
#include <bits/stdc++.h>
using namespace std;
#line 2 "template/macro.hpp"
#define rep(i, a, b) for (int i = (a); i < (int)(b); i++)
#define rrep(i, a, b) for (int i = (int)(b) - 1; i >= (a); i--)
#define ALL(v) (v).begin(), (v).end()
#define UNIQUE(v) sort(ALL(v)), (v).erase(unique(ALL(v)), (v).end())
#define SZ(v) (int)v.size()
#define MIN(v) *min_element(ALL(v))
#define MAX(v) *max_element(ALL(v))
#define LB(v, x) int(lower_bound(ALL(v), (x)) - (v).begin())
#define UB(v, x) int(upper_bound(ALL(v), (x)) - (v).begin())
#define YN(b) cout << ((b) ? "YES" : "NO") << "\n";
#define Yn(b) cout << ((b) ? "Yes" : "No") << "\n";
#define yn(b) cout << ((b) ? "yes" : "no") << "\n";
#line 6 "template/template.hpp"
#line 2 "template/util.hpp"
using uint = unsigned int;
using ll = long long int;
using ull = unsigned long long;
using i128 = __int128_t;
using u128 = __uint128_t;
template <class T, class S = T>
S SUM(const vector<T> &a) {
return accumulate(ALL(a), S(0));
}
template <class T>
inline bool chmin(T &a, T b) {
if (a > b) {
a = b;
return true;
}
return false;
}
template <class T>
inline bool chmax(T &a, T b) {
if (a < b) {
a = b;
return true;
}
return false;
}
template <class T>
int popcnt(T x) {
return __builtin_popcountll(x);
}
template <class T>
int topbit(T x) {
return (x == 0 ? -1 : 63 - __builtin_clzll(x));
}
template <class T>
int lowbit(T x) {
return (x == 0 ? -1 : __builtin_ctzll(x));
}
#line 8 "template/template.hpp"
#line 2 "template/inout.hpp"
struct Fast {
Fast() {
cin.tie(nullptr);
ios_base::sync_with_stdio(false);
cout << fixed << setprecision(15);
}
} fast;
template <class T1, class T2>
istream &operator>>(istream &is, pair<T1, T2> &p) {
return is >> p.first >> p.second;
}
template <class T1, class T2>
ostream &operator<<(ostream &os, const pair<T1, T2> &p) {
return os << p.first << " " << p.second;
}
template <class T>
istream &operator>>(istream &is, vector<T> &a) {
for (auto &v : a) is >> v;
return is;
}
template <class T>
ostream &operator<<(ostream &os, const vector<T> &a) {
for (auto it = a.begin(); it != a.end();) {
os << *it;
if (++it != a.end()) os << " ";
}
return os;
}
template <class T>
ostream &operator<<(ostream &os, const set<T> &st) {
os << "{";
for (auto it = st.begin(); it != st.end();) {
os << *it;
if (++it != st.end()) os << ",";
}
os << "}";
return os;
}
template <class T1, class T2>
ostream &operator<<(ostream &os, const map<T1, T2> &mp) {
os << "{";
for (auto it = mp.begin(); it != mp.end();) {
os << it->first << ":" << it->second;
if (++it != mp.end()) os << ",";
}
os << "}";
return os;
}
void in() {}
template <typename T, class... U>
void in(T &t, U &...u) {
cin >> t;
in(u...);
}
void out() { cout << "\n"; }
template <typename T, class... U, char sep = ' '>
void out(const T &t, const U &...u) {
cout << t;
if (sizeof...(u)) cout << sep;
out(u...);
}
#line 10 "template/template.hpp"
#line 2 "template/debug.hpp"
#ifdef LOCAL
#define debug 1
#define show(...) _show(0, #__VA_ARGS__, __VA_ARGS__)
#else
#define debug 0
#define show(...) true
#endif
template <class T>
void _show(int i, T name) {
cerr << '\n';
}
template <class T1, class T2, class... T3>
void _show(int i, const T1 &a, const T2 &b, const T3 &...c) {
for (; a[i] != ',' && a[i] != '\0'; i++) cerr << a[i];
cerr << ":" << b << " ";
_show(i + 1, a, c...);
}
#line 2 "modint/modint.hpp"
template <unsigned int m = 998244353>
struct ModInt {
using mint = ModInt;
unsigned int _v;
static constexpr unsigned int get_mod() { return m; }
static mint raw(int v) {
mint x;
x._v = v;
return x;
}
ModInt() : _v(0) {}
ModInt(int64_t v) {
long long x = (long long)(v % (long long)(umod()));
if (x < 0) x += umod();
_v = (unsigned int)(x);
}
unsigned int val() const { return _v; }
mint &operator++() {
_v++;
if (_v == umod()) _v = 0;
return *this;
}
mint &operator--() {
if (_v == 0) _v = umod();
_v--;
return *this;
}
mint operator++(int) {
mint result = *this;
++*this;
return result;
}
mint operator--(int) {
mint result = *this;
--*this;
return result;
}
mint &operator+=(const mint &rhs) {
_v += rhs._v;
if (_v >= umod()) _v -= umod();
return *this;
}
mint &operator-=(const mint &rhs) {
_v -= rhs._v;
if (_v >= umod()) _v += umod();
return *this;
}
mint &operator*=(const mint &rhs) {
unsigned long long z = _v;
z *= rhs._v;
_v = (unsigned int)(z % umod());
return *this;
}
mint &operator/=(const mint &rhs) { return *this = *this * rhs.inv(); }
mint operator+() const { return *this; }
mint operator-() const { return mint() - *this; }
mint pow(long long n) const {
assert(0 <= n);
mint x = *this, r = 1;
while (n) {
if (n & 1) r *= x;
x *= x;
n >>= 1;
}
return r;
}
mint inv() const {
assert(_v);
return pow(umod() - 2);
}
friend mint operator+(const mint &lhs, const mint &rhs) {
return mint(lhs) += rhs;
}
friend mint operator-(const mint &lhs, const mint &rhs) {
return mint(lhs) -= rhs;
}
friend mint operator*(const mint &lhs, const mint &rhs) {
return mint(lhs) *= rhs;
}
friend mint operator/(const mint &lhs, const mint &rhs) {
return mint(lhs) /= rhs;
}
friend bool operator==(const mint &lhs, const mint &rhs) {
return lhs._v == rhs._v;
}
friend bool operator!=(const mint &lhs, const mint &rhs) {
return lhs._v != rhs._v;
}
friend istream &operator>>(istream &is, mint &x) {
return is >> x._v;
}
friend ostream &operator<<(ostream &os, const mint &x) {
return os << x.val();
}
private:
static constexpr unsigned int umod() { return m; }
};
#line 5 "verify/fps/LC_sum_of_exponential_times_polynomial_limit.test.cpp"
using mint = ModInt<998244353>;
#line 2 "modint/factorial.hpp"
template <class mint>
struct Factorial {
static void reserve(int n) {
inv(n);
fact(n);
fact_inv(n);
}
static mint inv(int n) {
static long long mod = mint::get_mod();
static vector<mint> buf({0, 1});
assert(n != 0);
if (mod != mint::get_mod()) {
mod = mint::get_mod();
buf = vector<mint>({0, 1});
}
while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
while ((int)buf.size() <= n) {
long long k = buf.size(), q = (mod + k - 1) / k;
buf.push_back(q * buf[k * q - mod]);
}
return buf[n];
}
static mint fact(int n) {
static long long mod = mint::get_mod();
static vector<mint> buf({1, 1});
assert(n >= 0);
if (mod != mint::get_mod()) {
mod = mint::get_mod();
buf = vector<mint>({1, 1});
}
while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
while ((int)buf.size() <= n) {
long long k = buf.size();
buf.push_back(buf.back() * k);
}
return buf[n];
}
static mint fact_inv(int n) {
static long long mod = mint::get_mod();
static vector<mint> buf({1, 1});
assert(n >= 0);
if (mod != mint::get_mod()) {
mod = mint::get_mod();
buf = vector<mint>({1, 1});
}
if ((int)buf.size() <= n) inv(n);
while ((int)buf.capacity() <= n) buf.reserve(buf.capacity() * 2);
while ((int)buf.size() <= n) {
long long k = buf.size();
buf.push_back(buf.back() * inv(k));
}
return buf[n];
}
static mint binom(int n, int r) {
if (r < 0 || r > n) return 0;
return fact(n) * fact_inv(r) * fact_inv(n - r);
}
static mint binom_naive(int n, int r) {
if (r < 0 || r > n) return 0;
mint res = fact_inv(r);
for (int i = 0; i < r; i++) res *= n - i;
return res;
}
static mint multinom(const vector<int>& r) {
int n = 0;
for (auto& x : r) {
if (x < 0) return 0;
n += x;
}
mint res = fact(n);
for (auto& x : r) res *= fact_inv(x);
return res;
}
static mint P(int n, int r) {
if (r < 0 || r > n) return 0;
return fact(n) * fact_inv(n - r);
}
// partition n items to r groups (allow empty group)
static mint H(int n, int r) {
if (n < 0 || r < 0) return 0;
return r == 0 ? 1 : binom(n + r - 1, r);
}
};
/**
* @brief 階乗, 二項係数
*/
#line 2 "math/lpf-table.hpp"
vector<int> LPFTable(int n) {
vector<int> lpf(n + 1, 0);
iota(lpf.begin(), lpf.end(), 0);
for (int p = 2; p * p <= n; p += (p & 1) + 1) {
if (lpf[p] != p) continue;
for (int i = p * p; i <= n; i += p)
if (lpf[i] == i) lpf[i] = p;
}
return lpf;
}
/**
* @brief LPF Table
*/
#line 3 "modint/power-table.hpp"
// 0^k,1^k,2^k,...,n^k
template <class T>
vector<T> PowerTable(int n, int k) {
assert(k >= 0);
vector<T> f;
if (k == 0) {
f = vector<T>(n + 1, 0);
f[0] = 1;
} else {
f = vector<T>(n + 1, 1);
f[0] = 0;
auto lpf = LPFTable(n);
for (int i = 2; i <= n; i++)
f[i] = lpf[i] == i ? T(i).pow(k) : f[i / lpf[i]] * f[lpf[i]];
}
return f;
}
/**
* @brief Power Table
*/
#line 3 "fps/interpolate.hpp"
// f(0),f(1),...,f(n-1) -> f(x)
template <class mint>
mint Interpolate(const vector<mint>& f, mint x) {
int n = f.size();
vector<mint> l(n, 1), r(n, 1);
for (int i = 0; i + 1 < n; i++) l[i + 1] = l[i] * (x - i);
for (int i = n - 1; i > 0; i--) r[i - 1] = r[i] * (x - i);
using fact = Factorial<mint>;
mint s = 0;
for (int i = 0; i < n; i++) {
mint v = f[i] * l[i] * r[i] * fact::fact_inv(i) * fact::fact_inv(n - 1 - i);
if ((n - i) & 1)
s += v;
else
s -= v;
}
return s;
}
/**
* @brief Interpolate
* @docs docs/fps/interpolate.md
*/
#line 4 "fps/sum-of-exp-poly.hpp"
// sum_{i=0}^{infty}r^i*poly(i)
// f[i]=poly(i)
template <class mint>
mint SumOfExpPolyLimit(mint r, vector<mint>& f) {
if (r == 0) return f[0];
assert(r != 1);
int k = f.size();
vector<mint> g(k + 1, 0);
mint prod = 1;
for (int i = 0; i < k; i++) {
g[i + 1] = g[i] + f[i] * prod;
prod *= r;
}
using fact = Factorial<mint>;
mint c = 0;
prod = 1;
for (int i = 0; i <= k; i++) {
c += fact::binom(k, i) * prod * g[k - i];
prod *= -r;
}
c /= (1 - r).pow(k);
return c;
}
// sum_{i=0}^{n-1}r^i*poly(i)
// f[i]=poly(i)
template <class mint>
mint SumOfExpPoly(long long n, mint r, vector<mint>& f) {
if (n <= 0) return 0;
if (r == 0) return f[0];
int k = f.size();
vector<mint> g(k + 1, 0);
mint prod = 1;
for (int i = 0; i < k; i++) {
g[i + 1] = g[i] + f[i] * prod;
prod *= r;
}
if (r == 1) return Interpolate(g, mint(n));
mint c = 0;
prod = 1;
using fact = Factorial<mint>;
for (int i = 0; i <= k; i++) {
c += fact::binom(k, i) * prod * g[k - i];
prod *= -r;
}
c /= (1 - r).pow(k);
for (int i = 0; i <= k; i++) g[i] -= c;
mint ir = r.inv();
prod = 1;
for (int i = 1; i <= k; i++) g[i] *= (prod *= ir);
return Interpolate(g, mint(n)) * r.pow(n) + c;
}
/**
* @brief $\sum_{i}r^i poly(i)$
* @docs docs/fps/sum-of-exp-poly.md
*/
#line 9 "verify/fps/LC_sum_of_exponential_times_polynomial_limit.test.cpp"
int main() {
mint r;
int d;
in(r, d);
auto f = PowerTable<mint>(d, d);
out(SumOfExpPolyLimit(r, f));
}
verify/fps/LC_sum_of_exponential_times_polynomial_limit.test.cpp