#define AUTO_GENERATED
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial
#include "../../../src/comb/seq/pows/lib.hpp"
#include "../../../src/math/ipaf/sum/lib.hpp"
using namespace tifa_libs;
CEXP u32 MOD = 998244353;
#include "../../../src/math/ds/mint/md64/lib.hpp"
#include "../../../src/math/fact/helper/lib.hpp"
using namespace tifa_libs;
using mint = mint_md64<__LINE__>;
using namespace tifa_libs;
using fact_t = fact_helper<mint>;
int main() {
mint::set_mod(MOD);
std::cin.tie(nullptr)->std::ios::sync_with_stdio(false);
u32 r, d;
u64 n;
std::cin >> r >> d >> n;
auto p = tifa_libs::gen_pows<mint>(d + 1, d);
std::cout << tifa_libs::sum_ipaf<mint, fact_t>(p, mint(r), n);
return 0;
}
#line 1 "test/cpv/library-checker-other/sum_of_exponential_times_polynomial.mintd-md64.factorial-fact_helper.cpp"
#define AUTO_GENERATED
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/sum_of_exponential_times_polynomial
#line 2 "src/comb/seq/pows/lib.hpp"
#line 2 "src/math/qpow/mod/lib.hpp"
#line 2 "src/math/mul_mod/lib.hpp"
#line 2 "src/math/safe_mod/lib.hpp"
#line 2 "src/util/traits/math/lib.hpp"
// clang-format off
#line 2 "src/util/alias/num/lib.hpp"
#line 2 "src/util/util/lib.hpp"
// https://github.com/Tiphereth-A/CP-lib
#include <bits/extc++.h>
// clang-format off
namespace tifa_libs {
#define CEXP constexpr
#define CEXPE constexpr explicit
#define CR const&
#define CP const*
#define PC *const
#define CPC const*const
#define TPN typename
#define NE noexcept
#define CNE const noexcept
#define ND [[nodiscard]]
#define cT_(...) std::conditional_t<sizeof(__VA_ARGS__) <= sizeof(size_t) * 2, __VA_ARGS__, __VA_ARGS__ CR>
// NOLINTNEXTLINE(misc-const-correctness)
#define flt_(T, i, l, r, ...) for (T i = (l), i##e = (r)__VA_OPT__(, ) __VA_ARGS__; i < i##e; ++i)
#define retif_(cond, if_true, ...) if cond return if_true __VA_OPT__(; else return __VA_ARGS__)
#ifdef ONLINE_JUDGE
#undef assert
#define assert(x) 42
#endif
using namespace std::ranges;
using namespace std::literals;
template <class T>
CEXP T abs(T x) NE { retif_((x < 0), -x, x); }
} // namespace tifa_libs
// clang-format on
#line 4 "src/util/alias/num/lib.hpp"
// clang-format off
namespace tifa_libs {
#define mk0_(w, t) using w = t; using c##w = const t
#define mk_(w, t) mk0_(w, t); CEXP w operator""_##w(unsigned long long x) NE { return (w)x; }
mk_(i8, int8_t) mk_(u8, uint8_t) mk_(i16, int16_t) mk_(u16, uint16_t) mk_(i32, int32_t) mk_(u32, uint32_t) mk_(i64, int64_t) mk_(u64, uint64_t) mk_(isz, ssize_t) mk_(usz, size_t) mk_(chr, char) mk_(schr, signed char) mk_(uchr, unsigned char) mk_(sint, signed) mk_(uint, unsigned);
mk0_(i128, __int128_t); mk0_(u128, __uint128_t); mk0_(f32, float); mk0_(f64, double); mk0_(f128, long double);
#undef mk0_
#undef mk_
} // namespace tifa_libs
// clang-format on
#line 4 "src/util/traits/math/lib.hpp"
namespace tifa_libs {
template <class T> concept char_c = std::same_as<T, char> || std::same_as<T, signed char> || std::same_as<T, unsigned char>;
#pragma GCC diagnostic ignored "-Wpedantic"
template <class T> concept s128_c = std::same_as<T, __int128_t> || std::same_as<T, __int128>;
template <class T> concept u128_c = std::same_as<T, __uint128_t> || std::same_as<T, unsigned __int128>;
template <class T> concept i128_c = s128_c<T> || u128_c<T>;
#pragma GCC diagnostic warning "-Wpedantic"
template <class T> concept imost64_c = std::integral<T> && sizeof(T) * __CHAR_BIT__ <= 64;
template <class T> concept smost64_c = imost64_c<T> && std::signed_integral<T>;
template <class T> concept umost64_c = imost64_c<T> && std::unsigned_integral<T>;
template <class T> concept int_c = i128_c<T> || imost64_c<T>;
template <class T> concept sint_c = s128_c<T> || smost64_c<T>;
template <class T> concept uint_c = u128_c<T> || umost64_c<T>;
template <class T> concept arithm_c = std::is_arithmetic_v<T> || int_c<T>;
template <class T> concept mint_c = requires(T x) { {x.mod()} -> uint_c; {x.val()} -> uint_c; };
template <class T> concept dft_c = requires(T x, std::vector<TPN T::data_t> v, u32 n) { {x.size()} -> std::same_as<u32>; x.bzr(n); x.dif(v, n); x.dit(v, n); };
template <class T> concept ntt_c = dft_c<T> && requires(T x) { T::max_size; T::G; };
template <class T> struct to_sint : std::make_signed<T> {};
template <> struct to_sint<u128> { using type = i128; };
template <> struct to_sint<i128> { using type = i128; };
template <class T> using to_sint_t = TPN to_sint<T>::type;
template <class T> struct to_uint : std::make_unsigned<T> {};
template <> struct to_uint<u128> { using type = u128; };
template <> struct to_uint<i128> { using type = u128; };
template <class T> using to_uint_t = TPN to_uint<T>::type;
template <arithm_c T> struct to_bigger : std::make_unsigned<T> {};
#define _(w,ww) template <> struct to_bigger<w> { using type = ww; }
#define _2(w,ww) _(i##w,i##ww); _(u##w,u##ww);
_2(8, 16); _2(16, 32); _2(32, 64); _2(64, 128); _(f32, f64); _(f64, f128);
#undef _2
#undef _
template <class T> using to_bigger_t = TPN to_bigger<T>::type;
template <arithm_c T> CEXP T inf_v = [] {
if CEXP(sint_c<T>) return T(to_uint_t<T>(-1) / 4 - 1);
else if CEXP(uint_c<T>) return T(-1) / 2 - 1;
else return std::numeric_limits<T>::max() / 2 - 1;
}();
} // namespace tifa_libs
// clang-format on
#line 4 "src/math/safe_mod/lib.hpp"
namespace tifa_libs {
template <int_c T>
CEXP T safe_mod(T x, to_uint_t<T> mod) NE {
if CEXP (sint_c<T>) {
if (x <= -(T)mod || x >= (T)mod) x %= (T)mod;
retif_((x < 0), x + (T)mod, x);
} else {
retif_((x >= mod), x % mod, x);
}
}
} // namespace tifa_libs
#line 4 "src/math/mul_mod/lib.hpp"
namespace tifa_libs {
CEXP i64 mul_mod_s(i64 a, i64 b, u64 mod) NE {
if (std::bit_width((u64)abs(a)) + std::bit_width((u64)abs(b)) < 64) return safe_mod(a * b % (i64)mod, mod);
return safe_mod((i64)((i128)a * b % mod), mod);
}
CEXP u64 mul_mod_u(u64 a, u64 b, u64 mod) NE {
if (std::bit_width(a) + std::bit_width(b) <= 64) return a * b % mod;
return (u64)((u128)a * b % mod);
}
} // namespace tifa_libs
#line 4 "src/math/qpow/mod/lib.hpp"
namespace tifa_libs {
CEXP u64 qpow_mod(u64 a, u64 b, u64 mod) NE {
u64 res(1);
for (a %= mod; b; b >>= 1, a = mul_mod_u(a, a, mod)) {
while (!(b & 1)) b >>= 1, a = mul_mod_u(a, a, mod);
res = mul_mod_u(res, a, mod);
}
return res;
}
} // namespace tifa_libs
#line 2 "src/nt/lsieve/impl1/lib.hpp"
#line 2 "src/util/alias/others/lib.hpp"
#line 2 "src/util/consts/lib.hpp"
#line 4 "src/util/consts/lib.hpp"
// clang-format off
namespace tifa_libs {
using std::numbers::pi_v;
template <std::floating_point FP>
inline FP eps_v = std::sqrt(std::numeric_limits<FP>::epsilon());
template <std::floating_point FP>
CEXP void set_eps(FP v) NE { eps_v<FP> = v; }
CEXP u32 TIME = ((__TIME__[0] & 15) << 20) | ((__TIME__[1] & 15) << 16) | ((__TIME__[3] & 15) << 12) | ((__TIME__[4] & 15) << 8) | ((__TIME__[6] & 15) << 4) | (__TIME__[7] & 15);
CEXP auto STR2U16 = [] { std::array<u32, 65536> table{}; table.fill(-1_u32); flt_ (u32, i, 48, 58) flt_ (u32, j, 48, 58) table[i << 8 | j] = (j & 15) * 10 + (i & 15); return table; }();
inline const auto fn_0 = [](auto&&...) NE {};
inline const auto fn_is0 = [](auto x) NE { return x == 0; };
} // namespace tifa_libs
// clang-format on
#line 4 "src/util/alias/others/lib.hpp"
namespace tifa_libs {
template <class T>
struct chash {
CEXP static u64 C = u64(pi_v<f128> * 2e18) | 71;
CEXP u64 operator()(T x) CNE { return __builtin_bswap64(((u64)x ^ TIME) * C); }
};
// clang-format off
#define mk_(w, t) using w = t; using c##w = const t;
mk_(strn, std::string) mk_(strnv, std::string_view)
#undef mk_
template <class T> struct edge_t { T w; u32 u, v; CEXP auto operator<=>(edge_t CR) const = default; }; template <class T> using cedge_t = const edge_t<T>;
template <class T> struct pt3 { T _0, _1, _2; CEXP auto operator<=>(pt3 CR) const = default; }; template <class T> using cpt3 = const pt3<T>;
template <class T> struct pt4 { T _0, _1, _2, _3; CEXP auto operator<=>(pt4 CR) const = default; }; template <class T> using cpt4 = const pt4<T>;
#define mkT_(w, t, ...) template <class T> using w = t __VA_OPT__(, ) __VA_ARGS__; template <class T> using c##w = const t __VA_OPT__(, ) __VA_ARGS__;
mkT_(ptt, std::pair<T, T>) mkT_(alc, std::pmr::polymorphic_allocator<T>) mkT_(vec, std::vector<T>) mkT_(vvec, vec<vec<T>>) mkT_(v3ec, vvec<vec<T>>) mkT_(vecpt, vec<ptt<T>>) mkT_(vvecpt, vvec<ptt<T>>) mkT_(ptvec, ptt<vec<T>>) mkT_(ptvvec, ptt<vvec<T>>)
#undef mkT_
template <class T> using itl = std ::initializer_list<T>;
template <class T, usz ext = std::dynamic_extent> using spn = std::span<T const, ext>;
template <class T, usz N> using arr = std::array<T, N>; template <class T, usz N> using carr = std::array<const T, N>;
template <class U, class T> using vecp = vec<std::pair<U, T>>; template <class U, class T> using vvecp = vvec<std::pair<U, T>>;
template <class U, class T> using vvecp = vvec<std::pair<U, T>>; template <class U, class T> using vvvecp = vvec<vvec<std::pair<U, T>>>;
#ifdef PB_DS_ASSOC_CNTNR_HPP
template <class T, class C = std::less<T>> using set = __gnu_pbds::tree<T, __gnu_pbds::null_type, C>;
template <class K, class V, class C = std::less<K>> using map = __gnu_pbds::tree<K, V, C>;
// hset<u64> s({}, {}, {}, {}, {1<<16});
template <class T, class HF = chash<T>> using hset = __gnu_pbds::gp_hash_table<T, __gnu_pbds::null_type, HF>;
// hmap<u64, int> s({}, {}, {}, {}, {1<<16});
template <class K, class V, class HF = chash<K>> using hmap = __gnu_pbds::gp_hash_table<K, V, HF>;
#else
using std::set, std::map;
template <class T, class HF = chash<T>> using hset = std::unordered_set<T, HF>;
template <class K, class V, class HF = chash<K>> using hmap = std::unordered_map<K, V, HF>;
#endif
#ifdef PB_DS_PRIORITY_QUEUE_HPP
template <class T, class C = std::less<T>> using pq = __gnu_pbds::priority_queue<T, C>;
#else
template <class T, class C = std::less<T>> using pq = std::priority_queue<T, vec<T>, C>;
#endif
template <class T> using pqg = pq<T, std::greater<T>>;
// clang-format on
#define mk1_(V, A, T) using V##A = V<T>;
#define mk_(V, A, T) mk1_(V, A, T) mk1_(c##V, A, T)
#define mk(A, T) mk_(edge_t, A, T) mk_(ptt, A, T) mk_(pt3, A, T) mk_(pt4, A, T) mk_(vec, A, T) mk_(vvec, A, T) mk_(v3ec, A, T) mk_(vecpt, A, T) mk_(vvecpt, A, T) mk_(ptvec, A, T) mk_(ptvvec, A, T) mk1_(spn, A, T) mk1_(itl, A, T)
mk(b, bool) mk(c, chr) mk(i, i32) mk(u, u32) mk(ii, i64) mk(uu, u64) mk(t, isz) mk(z, usz) mk(f, f32) mk(d, f64) mk(s, strn);
#undef mk
#undef mk_
#undef mk1_
} // namespace tifa_libs
#line 4 "src/nt/lsieve/impl1/lib.hpp"
namespace tifa_libs {
template <class... Ts>
struct lsieve : Ts... {
vecb not_prime;
vecu primes;
CEXPE lsieve(u32 n) NE : Ts(n)..., not_prime(n) {
if (n < 2) return;
// clang-format off
primes.reserve((usz)max(127, int(n * (n >= 2e5 ? 1.6 : 1.7) / std::bit_width(n) + 1)));
// clang-format on
flt_ (u32, i, 2, n) {
if (!not_prime[i]) primes.push_back(i), (Ts::prime(i), ...);
for (auto j : primes) {
if (i * j >= n) break;
not_prime[i * j] = true;
if (i % j) (Ts::coprime(i, j), ...);
else {
(Ts::not_coprime(i, j), ...);
break;
}
}
}
primes.shrink_to_fit();
}
};
} // namespace tifa_libs
#line 5 "src/comb/seq/pows/lib.hpp"
namespace tifa_libs {
namespace gen_pows_impl_ {
struct ls_pows {
static inline u64 b, mod;
vecuu pows;
protected:
CEXPE ls_pows(u32 n) NE : pows(n) {
if (n > 1) pows[1] = 1;
}
void prime(u32 p) NE { pows[p] = qpow_mod(p, b, mod); }
void coprime(u32 i, u32 j) NE { pows[i * j] = mul_mod_u(pows[i], pows[j], mod); }
void not_coprime(u32 i, u32 j) NE { coprime(i, j); }
};
} // namespace gen_pows_impl_
// i^{b} from i=0..n-1
CEXP vecuu gen_pows(u32 n, u64 b, u64 mod) NE {
retif_((!b) [[unlikely]], vecuu(n, mod > 1));
retif_((!n) [[unlikely]], {});
gen_pows_impl_::ls_pows::b = b;
gen_pows_impl_::ls_pows::mod = mod;
return lsieve<gen_pows_impl_::ls_pows>(n).pows;
}
// i^{b} from i=0..n-1
template <class mint>
CEXP vec<mint> gen_pows(u32 n, u64 b) NE {
vec<mint> ans(n);
auto _ = gen_pows(n, b, mint::mod());
flt_ (u32, i, 0, n) ans[i] = _[i];
return ans;
}
} // namespace tifa_libs
#line 2 "src/math/ipaf/sum/lib.hpp"
#line 2 "src/comb/binom/lib.hpp"
#line 2 "src/math/fact/helper/lib.hpp"
#line 5 "src/math/fact/helper/lib.hpp"
namespace tifa_libs {
template <mint_c mint>
struct fact_helper {
using val_t = mint;
static CEXP u32 DEFAULT_MAX = 10'000'001;
static CEXP u64 mod() NE { return val_t::mod(); }
static inline vec<val_t> fact, ifact;
fact_helper() = delete;
// ensure fact.size() >= sz
static CEXP void ensure(u32 sz = DEFAULT_MAX) NE {
if (sz = max(2_u32, min((u32)mod(), sz)); sz <= fact.size()) return;
u32 pre = (u32)fact.size();
fact.resize(sz), ifact.resize(sz);
if (pre < 2) pre = 2, fact[0] = fact[1] = ifact[0] = ifact[1] = 1;
flt_ (u32, i, pre, sz) fact[i] = fact[i - 1] * i;
ifact.back() = fact.back().inv();
for (u32 i = sz - 1; i > pre; --i) ifact[i - 1] = ifact[i] * i;
}
static CEXP val_t get_fact(u64 n) NE {
if (n >= mod()) [[unlikely]]
return 0;
if (fact.empty()) [[unlikely]]
ensure();
if (n < fact.size()) [[likely]]
return fact[n];
val_t _ = fact.back() * n;
flt_ (u64, i, fact.size(), n) _ *= i;
return _;
}
static CEXP val_t get_ifact(u64 n) NE {
if (n >= mod()) [[unlikely]]
return 0;
if (fact.empty()) [[unlikely]]
ensure();
if (n < ifact.size()) [[likely]]
return ifact[n];
return get_fact(n).inv();
}
};
} // namespace tifa_libs
#line 4 "src/comb/binom/lib.hpp"
namespace tifa_libs {
template <class mint, class fact = fact_helper<mint>>
requires std::same_as<mint, TPN fact::val_t>
struct binom {
using fact_t = fact;
CEXPE binom(u32 max_m = fact::DEFAULT_MAX) NE { fact::ensure(max_m + 1); }
// $\binom{m}{n}$
CEXP mint mCn(uint_c auto m, uint_c auto n) CNE { retif_((m < n) [[unlikely]], 0, mPn(m, n) * fact::get_ifact(n)); }
// $\binom{m}{n}$
template <sint_c T>
CEXP mint mCn(T m, T n) CNE { retif_((m < n || n < 0) [[unlikely]], 0, mCn(to_uint_t<T>(m), to_uint_t<T>(n))); }
//! mint::mod() must be prime
template <int_c T>
CEXP mint lucas(T m, T n) CNE {
assert(mint::mod() > 1);
auto f = [this](auto&& f, auto m, auto n) NE -> mint { retif_((n == 0), 1, this->mCn(m % fact::mod(), n % fact::mod()) * f(f, m / fact::mod(), n / fact::mod())); };
retif_((m < n || n < 0) [[unlikely]], 0, f(f, to_uint_t<T>(m), to_uint_t<T>(n)));
}
// $\binom{m}{n} \cdot n!$
CEXP mint mPn(uint_c auto m, uint_c auto n) CNE { retif_((m < n) [[unlikely]], 0, fact::get_fact(m) * fact::get_ifact(m - n)); }
// $\binom{m}{n} \cdot n!$
template <sint_c T>
CEXP mint mPn(T m, T n) CNE { retif_((m < n || n < 0) [[unlikely]], 0, mPn(to_uint_t<T>(m), to_uint_t<T>(n))); }
// $[x^n] \frac{1}{(1-x)^m}$
CEXP mint mHn(uint_c auto m, uint_c auto n) CNE { retif_((n <= 0), n == 0, mCn(m + n - 1, n)); }
// $[x^n] \frac{1}{(1-x)^m}$
template <sint_c T>
CEXP mint mHn(T m, T n) CNE { retif_((m < 0 || n <= 0), n == 0, mHn(to_uint_t<T>(m), to_uint_t<T>(n))); }
};
} // namespace tifa_libs
#line 2 "src/math/qpow/basic/lib.hpp"
#line 4 "src/math/qpow/basic/lib.hpp"
namespace tifa_libs {
template <class T>
CEXP T qpow(T a, u64 b, cT_(T) init_v = T{1}) NE {
T res = init_v;
for (; b; b >>= 1, a = a * a) {
while (!(b & 1)) b >>= 1, a = a * a;
res = res * a;
}
return res;
}
} // namespace tifa_libs
#line 2 "src/math/interp/lagrange0/lib.hpp"
#line 2 "src/comb/seq/ifact/lib.hpp"
#line 2 "src/comb/seq/inv/lib.hpp"
#line 5 "src/comb/seq/inv/lib.hpp"
namespace tifa_libs {
// i^{-1} from i=0..n-1
CEXP vecuu gen_inv(u32 n, u64 mod) NE {
retif_((n <= 1) [[unlikely]], vecuu(n, 1));
vecuu ans(n);
ans[0] = ans[1] = 1;
flt_ (u32, i, 2, n) ans[i] = mul_mod_u(mod - mod / i, ans[mod % i], mod);
return ans;
}
// i^{-1} from i=0..n-1
template <class mint>
CEXP vec<mint> gen_inv(u32 n) NE {
vec<mint> ans(n);
auto _ = gen_inv(n, mint::mod());
flt_ (u32, i, 0, n) ans[i] = _[i];
return ans;
}
} // namespace tifa_libs
#line 5 "src/comb/seq/ifact/lib.hpp"
namespace tifa_libs {
// (i!)^{-1} from i=0..n-1
CEXP vecuu gen_ifact(u32 n, u64 mod, vecuu inv) NE {
flt_ (u32, i, 2, n) inv[i] = mul_mod_u(inv[i], inv[i - 1], mod);
return inv;
}
// (i!)^{-1} from i=0..n-1
CEXP vecuu gen_ifact(u32 n, u64 mod) NE { return gen_ifact(n, mod, gen_inv(n, mod)); }
// (i!)^{-1} from i=0..n-1
template <class mint>
CEXP vec<mint> gen_ifact(u32 n, vec<mint> inv) NE {
flt_ (u32, i, 2, n) inv[i] *= inv[i - 1];
return inv;
}
// (i!)^{-1} from i=0..n-1
template <class mint>
CEXP vec<mint> gen_ifact(u32 n) NE { return gen_ifact(n, gen_inv<mint>(n)); }
} // namespace tifa_libs
#line 5 "src/math/interp/lagrange0/lib.hpp"
namespace tifa_libs {
CEXP i64 lagrange_interp0(spnii v, u64 x, u64 mod, spnuu ifact) NE {
cu32 n = (u32)v.size();
retif_((assert(n); n == 1) [[unlikely]], v[0]);
if (x < n) return v[x];
vecuu pre(n);
flt_ (u32, i, 0, n) pre[i] = x - i;
flt_ (u32, i, 1, n) pre[i] = mul_mod_u(pre[i], pre[i - 1], mod);
vecuu suc(n);
flt_ (u32, i, 0, n) suc[i] = x - i;
for (u32 i = n - 2; ~i; --i) suc[i] = mul_mod_u(suc[i], suc[i + 1], mod);
i64 ans = 0;
flt_ (u32, i, 0, n) {
i64 _ = v[i];
if (i) _ = mul_mod_s(_, (i64)pre[i - 1], mod);
if (i + 1 < n) _ = mul_mod_s(_, (i64)suc[i + 1], mod);
_ = mul_mod_s(mul_mod_s(_, (i64)ifact[i], mod), (i64)ifact[n - i - 1], mod);
ans = (ans + ((n - i) % 2 ? _ : (i64)mod - _)) % (i64)mod;
}
return ans;
}
CEXP i64 lagrange_interp0(spnii v, u64 x, u64 mod) NE { return lagrange_interp0(v, x, mod, gen_ifact((u32)v.size(), mod)); }
template <class mint>
CEXP mint lagrange_interp0(vec<mint> CR v, u64 x, vec<mint> CR ifact) NE {
vecii _(v.size());
flt_ (u32, i, 0, (u32)v.size()) _[i] = (i64)v[i].val();
vecuu ifa(ifact.size());
flt_ (u32, i, 0, (u32)ifact.size()) ifa[i] = ifact[i].val();
return mint(lagrange_interp0(_, x, mint::mod(), ifa));
}
template <class mint>
CEXP mint lagrange_interp0(vec<mint> CR v, u64 x) NE { return lagrange_interp0(v, x, mint::mod(), gen_ifact<mint>(v.size())); }
} // namespace tifa_libs
#line 6 "src/math/ipaf/sum/lib.hpp"
namespace tifa_libs {
// @param f $f(0),\dots,f(k-1)$, $k\leq n$
// @return $\sum_{i=0}^{n-1}a^if(i)$
template <class mint, class fact>
CEXP mint sum_ipaf(vec<mint> CR f, cT_(mint) a, u64 n, binom<mint, fact> CR C) NE {
using fact_t = TPN binom<mint>::fact_t;
retif_((!n) [[unlikely]], mint(0));
if (!a.val()) return f[0];
if (a.val() == 1) {
vec<mint> g(f.size() + 1, mint(0));
flt_ (u32, i, 1, (u32)g.size()) g[i] = g[i - 1] + f[i - 1];
return lagrange_interp0(g, n, fact_t::ifact);
}
vec<mint> g(f.size());
mint _0 = 1;
flt_ (u32, i, 0, (u32)g.size()) g[i] = f[i] * _0, _0 *= a;
flt_ (u32, i, 1, (u32)g.size()) g[i] += g[i - 1];
mint c = 0, _1 = 1;
cu32 K = u32(f.size() - 1);
flt_ (u32, i, 0, K + 1) c += C.mCn(K + 1, i) * _1 * g[K - i], _1 *= -a;
c /= qpow(-a + 1, K + 1);
mint _2 = 1, ia = a.inv();
flt_ (u32, i, 0, (u32)g.size()) g[i] = (g[i] - c) * _2, _2 *= ia;
return lagrange_interp0(g, n - 1, fact_t::ifact) * qpow(a, n - 1) + c;
}
template <class mint, class fact = fact_helper<mint>>
CEXP mint sum_ipaf(vec<mint> CR f, cT_(mint) a, u64 n) NE {
retif_((!n) [[unlikely]], mint(0));
if (!a.val()) return f[0];
return sum_ipaf(f, a, n, binom<mint, fact>((u32)(f.size() + 1)));
}
} // namespace tifa_libs
#line 5 "test/cpv/library-checker-other/sum_of_exponential_times_polynomial.mintd-md64.factorial-fact_helper.cpp"
using namespace tifa_libs;
CEXP u32 MOD = 998244353;
#line 2 "src/math/ds/mint/md64/lib.hpp"
#line 2 "src/nt/mod/montgomery64/lib.hpp"
#line 4 "src/nt/mod/montgomery64/lib.hpp"
namespace tifa_libs {
template <u64 MOD>
struct montgomery64 {
static CEXP u64 R = [] {
u64 iv = MOD * (2 - MOD * MOD);
iv *= 2 - MOD * iv, iv *= 2 - MOD * iv, iv *= 2 - MOD * iv;
return iv * (2 - MOD * iv);
}();
static CEXP u64 R2 = [] {
u64 iv = -MOD % MOD;
for (u32 i = 0; i != 64; ++i)
if ((iv *= 2) >= MOD) iv -= MOD;
return iv;
}();
static_assert(MOD & 1);
static_assert(R * MOD == 1);
static_assert((MOD >> 63) == 0);
static_assert(MOD != 1);
static CEXP u64 mulh(u64 x, u64 y) NE { return u64((u128)x * y >> 64); }
static CEXP u64 redc_mul(u64 x, u64 y) NE {
u64 res = mulh(x, y) - mulh(x * y * R, MOD);
return res + (MOD & -(res >> 63));
}
static CEXP u64 norm(i64 x) NE { return (u64)x + (MOD & u64(-(x < 0))); }
};
template <> // dynamic
struct montgomery64<0> {
u64 MOD, R, R2;
CEXP montgomery64() NE = default;
CEXPE montgomery64(u64 m) NE { reset(m); }
CEXP void reset(u64 m) NE {
assert(!((m & 1) == 0 || m == 1 || m >> 63)), MOD = m;
u64 iv = MOD * (2 - MOD * MOD);
iv *= 2 - MOD * iv, iv *= 2 - MOD * iv, iv *= 2 - MOD * iv, R = iv * (2 - MOD * iv), R2 = -MOD % MOD;
flt_ (u32, i, 0, 64)
if ((R2 *= 2) >= MOD) R2 -= MOD;
}
ND CEXP u64 mul_h(u64 x, u64 y) CNE { return u64((u128)x * y >> 64); }
ND CEXP u64 redc_mul(u64 x, u64 y) CNE {
cu64 res = mul_h(x, y) - mul_h(x * y * R, MOD);
return res + (MOD & -(res >> 63));
}
ND CEXP u64 norm(i64 x) CNE { return u64(x + i64(MOD & u64(-(x < 0)))); }
};
} // namespace tifa_libs
#line 2 "src/math/ds/mint/_base/lib.hpp"
#line 2 "src/nt/inverse/lib.hpp"
#line 2 "src/nt/gl/inv_gcd/lib.hpp"
#line 2 "src/nt/gl/exgcd/lib.hpp"
#line 4 "src/nt/gl/exgcd/lib.hpp"
namespace tifa_libs {
// Binary exgcd
template <uint_c U, bool only_x = false>
CEXP auto exgcd_b(U a, U b) NE {
using T = to_sint_t<U>;
if CEXP (only_x) {
if (!a) return std::make_tuple(b, (T)0);
if (!b) return std::make_tuple(a, (T)1);
} else {
if (!a) return std::make_tuple(b, (T)0, (T) !!b);
if (!b) return std::make_tuple(a, (T)1, (T)0);
}
auto r = std::__countr_zero(a | b);
a >>= r, b >>= r;
T x = (T)a, y = (T)b, s = 1, t = 0, u = 0, v = 1;
while (x) {
while (!(x & 1))
if (x /= 2; !((s | t) & 1)) s /= 2, t /= 2;
else s = (s + (T)b) / 2, t = (t - (T)a) / 2;
while (!(y & 1))
if (y /= 2; !((u | v) & 1)) u /= 2, v /= 2;
else u = (u + (T)b) / 2, v = (v - (T)a) / 2;
if (x >= y) x -= y, s -= u, t -= v;
else y -= x, u -= s, v -= t;
}
if (y > 1) a /= (U)y, b /= (U)y;
if (a && (U)abs(v) >= a) {
const T _ = v / (T)a;
v -= _ * (T)a, u += _ * (T)b;
}
if (b && (U)abs(u) >= b) {
const T _ = u / (T)b;
u -= _ * (T)b, v += _ * (T)a;
}
if (const T u_ = u + (T)b, v_ = v - (T)a; abs(u_) + abs(v_) <= abs(u) + abs(v)) u = u_, v = v_;
if (const T u_ = u - (T)b, v_ = v + (T)a; abs(u_) + abs(v_) <= abs(u) + abs(v)) u = u_, v = v_;
if CEXP (only_x) return std::make_tuple(U(y << r), u);
else return std::make_tuple(U(y << r), u, v);
}
// @return then return tuple(g, x[, y]) s.t. g = gcd(a, b), xa + yb = g, |x| + |y| is the minimal (primary) and x <= y (secondarily)
template <sint_c T, bool only_x = false>
CEXP auto exgcd(T a, T b) NE {
using U = to_uint_t<T>;
if (auto [x, y] = minmax(a, b); x >= 0 && y <= T(U(-1) >> sizeof(U))) return exgcd_b<U, only_x>((U)a, (U)b);
if CEXP (only_x) {
T s = 1, u = 0;
while (b) {
T c = a / b;
std::tie(s, u, a, b) = std::make_tuple(u, s - u * c, b, a - b * c);
}
return std::make_tuple((U)a, s);
} else {
T s = 1, t = 0, u = 0, v = 1;
while (b) {
T c = a / b;
std::tie(s, t, u, v, a, b) = std::make_tuple(u, v, s - u * c, t - v * c, b, a - b * c);
}
return std::make_tuple((U)a, s, t);
}
}
} // namespace tifa_libs
#line 6 "src/nt/gl/inv_gcd/lib.hpp"
namespace tifa_libs {
template <uint_c T>
CEXP ptt<T> inv_gcd(T n, T mod) NE {
using U = to_sint_t<T>;
auto [g, x] = exgcd<U, true>(U(n % mod), (U)mod);
return {g, safe_mod(x, mod)};
}
} // namespace tifa_libs
#line 4 "src/nt/inverse/lib.hpp"
namespace tifa_libs {
// simple but slower: inv(n, mod) -> 1 < n ? mod - inv(mod % n, n) * mod / n : 1;
template <uint_c T, uint_c U>
CEXP U inverse(T n, U mod) NE {
auto [g, x] = inv_gcd(U(n % mod), mod);
assert(g == 1);
return x;
}
} // namespace tifa_libs
#line 2 "src/util/traits/others/lib.hpp"
// clang-format off
#line 4 "src/util/traits/others/lib.hpp"
namespace tifa_libs {
//! only for template without non-type argument
template <class, template <class...> class> CEXP bool specialized_from_v = false;
template <template <class...> class T, class... Args> CEXP bool specialized_from_v<T<Args...>, T> = true;
static_assert(specialized_from_v<vecu, std::vector>);
template <class T> concept container_c = common_range<T> && !std::is_array_v<std::remove_cvref_t<T>> && !std::same_as<std::remove_cvref_t<T>, strn> && !std::same_as<std::remove_cvref_t<T>, strnv>;
template <class T> concept istream_c = std::derived_from<T, std::istream> || std::derived_from<T, std::wistream> || requires(T is) { is.peek(); };
template <class T> concept ostream_c = std::derived_from<T, std::ostream> || std::derived_from<T, std::wostream> || requires(T os) { os.flush(); };
} // namespace tifa_libs
// clang-format on
#line 5 "src/math/ds/mint/_base/lib.hpp"
namespace tifa_libs::mint_impl_ {
struct mint_tag_base {};
template <std::derived_from<mint_tag_base> tag_t>
struct mint : tag_t {
CEXP mint() = default;
CEXP mint(int_c auto v) NE : tag_t(v) {}
using raw_t = tag_t::raw_t;
using sraw_t = to_sint_t<raw_t>;
static CEXP sraw_t smod() NE { return (sraw_t)tag_t::mod(); }
ND CEXP sraw_t sval() CNE { return (sraw_t)tag_t::val(); }
template <int_c T>
CEXPE operator T() CNE { return (T)tag_t::val(); }
CEXP mint& operator+=(mint CR r) NE {
mint::add(r);
return *this;
}
CEXP mint& operator-=(mint CR r) NE {
mint::sub(r);
return *this;
}
CEXP mint& operator*=(mint CR r) NE {
mint::mul(r);
return *this;
}
CEXP mint& operator/=(mint CR r) NE { return *this = *this * r.inv(); }
CEXP mint CR operator+() CNE { return *this; }
CEXP mint operator-() CNE { return tag_t::template neg<mint>(); }
ND CEXP mint inv() CNE { return inverse(tag_t::val(), tag_t::mod()); }
friend CEXP mint operator+(mint l, mint CR r) NE { return l += r; }
friend CEXP mint operator-(mint l, mint CR r) NE { return l -= r; }
friend CEXP mint operator*(mint l, mint CR r) NE { return l *= r; }
friend CEXP mint operator/(mint l, mint CR r) NE { return l /= r; }
friend CEXP bool operator==(mint CR l, mint CR r) NE { return l.val() == r.val(); }
friend CEXP auto operator<=>(mint CR l, mint CR r) NE { return l.sval() <=> r.sval(); }
friend auto& operator>>(istream_c auto& is, mint& x) NE {
i64 _;
is >> _, x = mint(_);
return is;
}
friend auto& operator<<(ostream_c auto& os, mint CR x) NE { return os << x.val(); }
friend CEXP auto abs(mint CR x) NE { return x.val(); }
};
} // namespace tifa_libs::mint_impl_
#line 5 "src/math/ds/mint/md64/lib.hpp"
namespace tifa_libs {
template <i64 ID>
class mint_md64_tag : public mint_impl_::mint_tag_base {
static inline montgomery64<0> core;
public:
static CEXP bool FIXED_MOD = false;
static CEXP void set_mod(u64 m) NE { core.reset(m); }
protected:
using raw_t = u64;
raw_t v_{};
CEXP mint_md64_tag() NE = default;
CEXP mint_md64_tag(int_c auto v) NE : v_{mod(v)} {}
public:
static CEXP raw_t mod(sint_c auto v) NE { retif_((v >= 0) [[likely]], mod((to_uint_t<decltype(v)>)v), core.redc_mul(core.norm(i64(v % (i64)mod())), core.R2)); }
static CEXP raw_t mod(uint_c auto v) NE {
if CEXP (umost64_c<decltype(v)>) {
retif_((cu64 x = (u64)v; x < mod()) [[likely]], core.redc_mul(x, core.R2), core.redc_mul(x % mod(), core.R2));
} else retif_((v < mod()) [[likely]], core.redc_mul((raw_t)v, core.R2), core.redc_mul((raw_t)(v % mod()), core.R2));
}
static CEXP raw_t mod() NE { return core.MOD; }
ND CEXP raw_t val() CNE {
const raw_t res = -core.mul_h(v_ * core.R, mod());
return res + (mod() & -(res >> 63));
}
CEXP raw_t& data() NE { return v_; }
protected:
template <class mint>
CEXP auto neg() CNE {
mint res;
res.v_ = (mod() & -raw_t(v_ != 0)) - v_;
return res;
}
CEXP void add(mint_md64_tag CR r) NE { v_ += r.v_ - mod(), v_ += mod() & -(v_ >> 63); }
CEXP void sub(mint_md64_tag CR r) NE { v_ -= r.v_, v_ += mod() & -(v_ >> 63); }
CEXP void mul(mint_md64_tag CR r) NE { v_ = core.redc_mul(v_, r.v_); }
};
template <i64 ID>
using mint_md64 = mint_impl_::mint<mint_md64_tag<ID>>;
} // namespace tifa_libs
#line 11 "test/cpv/library-checker-other/sum_of_exponential_times_polynomial.mintd-md64.factorial-fact_helper.cpp"
using namespace tifa_libs;
using mint = mint_md64<__LINE__>;
using namespace tifa_libs;
using fact_t = fact_helper<mint>;
int main() {
mint::set_mod(MOD);
std::cin.tie(nullptr)->std::ios::sync_with_stdio(false);
u32 r, d;
u64 n;
std::cin >> r >> d >> n;
auto p = tifa_libs::gen_pows<mint>(d + 1, d);
std::cout << tifa_libs::sum_ipaf<mint, fact_t>(p, mint(r), n);
return 0;
}
test/cpv/library-checker-other/sum_of_exponential_times_polynomial.mintd-md64.factorial-fact_helper.cpp