#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/fps/ds/ntt3/lib.hpp"
#include "../../../src/math/ds/mint/bs/lib.hpp"
#include "../../../src/math/ds/mint/md64/lib.hpp"
#include "../../../src/math/fact/helper_l/lib.hpp"
using namespace tifa_libs;
using mint = mint_md64<__LINE__>;
using namespace tifa_libs;
using mint_p3ntt1 = mint_bs<167772161>;
using mint_p3ntt2 = mint_bs<469762049>;
using mint_p3ntt3 = mint_bs<754974721>;
using poly = poly3ntt<mint, mint_p3ntt1, mint_p3ntt2, mint_p3ntt3>;
using namespace tifa_libs;
using fact_t = factl_helper<poly>;
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.mints-bs.poly_anymod-p3ntt.factorial-factl_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.mints-bs.poly_anymod-p3ntt.factorial-factl_helper.cpp"
using namespace tifa_libs;
CEXP u32 MOD = 998244353;
#line 2 "src/fps/ds/ntt3/lib.hpp"
#line 2 "src/conv/add/ntt3/lib.hpp"
#line 2 "src/nt/mod/barrett/lib.hpp"
#line 4 "src/nt/mod/barrett/lib.hpp"
namespace tifa_libs {
template <u64 MOD, u64 B_ = 1>
struct barrett {
static CEXP u64 B = B_ % MOD, R = ((u128)B << 64) / MOD;
static CEXP u64 reduce(u64 a) NE {
if (u64 q = u64((u128)a * R >> 64); (a = a * B - q * MOD) >= MOD) a -= MOD;
return a;
}
};
template <> // dynamic
struct barrett<0> {
u64 mod, b, r;
CEXP barrett() NE = default;
CEXPE barrett(u64 mod, u64 b = 1) NE { reset(mod, b); }
CEXP void reset(u64 mod_, u64 b_ = 1) NE { assert(mod_), mod = mod_, b = b_ % mod, r = (u64(((u128)b << 64) / mod)); }
ND CEXP u64 reduce(u64 a) CNE {
if (cu64 q = u64((u128)a * r >> 64); (a = a * b - q * mod) >= mod) a -= mod;
return a;
}
};
} // namespace tifa_libs
#line 2 "src/conv/trans/ntt/lib.hpp"
#line 2 "src/nt/proot/uint/lib.hpp"
#line 2 "src/nt/pfactors/lib.hpp"
#line 2 "src/edh/discretization/lib.hpp"
#line 2 "src/fast/rsort32/lib.hpp"
#line 4 "src/fast/rsort32/lib.hpp"
namespace tifa_libs {
template <class C>
requires(std::is_array_v<C> && std::integral<decltype(std::declval<C>()[0])> && sizeof(std::declval<C>()[0]) == 4) || (std::contiguous_iterator<TPN C::iterator> && std::integral<TPN C::value_type> && sizeof(TPN C::value_type) == 4)
void rsort32(C& a) NE {
if (a.size() <= 1) return;
if (a.size() <= 200'000) {
std::ranges::sort(a);
return;
}
arr<u32, 256> _0{}, _1{}, _2{}, _3{};
cu32 n = (u32)a.size();
vecu b(n);
u32 *a_ = (u32*)a.data(), *b_ = (u32*)b.data();
for (cu32 *_ = a_ + n, *i = a_; i < _; ++i) ++_0[*i & 255], ++_1[*i >> 8 & 255], ++_2[*i >> 16 & 255], ++_3[*i >> 24 & 255];
flt_ (u32, i, 1, 256) _0[i] += _0[i - 1], _1[i] += _1[i - 1], _2[i] += _2[i - 1], _3[i] += _3[i - 1];
for (u32 CP i = a_ + n; --i >= a_;) b_[--_0[*i & 255]] = *i;
for (u32 CP i = b_ + n; --i >= b_;) a_[--_1[*i >> 8 & 255]] = *i;
for (u32 CP i = a_ + n; --i >= a_;) b_[--_2[*i >> 16 & 255]] = *i;
for (u32 CP i = b_ + n; --i >= b_;) a_[--_3[*i >> 24 & 255]] = *i;
if CEXP (std::is_signed_v<TPN C::value_type>) {
u32 i = n;
while (i && a[i - 1] < 0) --i;
rotate(a_, a_ + n, a_ + i);
}
}
template <class C>
requires(std::is_array_v<C> && std::integral<decltype(std::declval<C>()[0])> && sizeof(std::declval<C>()[0]) == 4) || range<C>
void sort(C& a) NE {
if CEXP (std::is_array_v<C> || (std::contiguous_iterator<TPN C::iterator> && std::integral<TPN C::value_type> && sizeof(TPN C::value_type) == 4)) rsort32(a);
else std::ranges::sort(a);
}
} // namespace tifa_libs
#line 4 "src/edh/discretization/lib.hpp"
namespace tifa_libs {
template <common_range T>
CEXP T uniq(T v) NE {
tifa_libs::sort(v);
auto r = unique(begin(v), end(v));
return {begin(v), begin(r)};
}
template <common_range T>
CEXP std::pair<T, vecu> gen_id(T CR v) NE {
const T _ = uniq(v);
vecu _1;
_1.reserve(v.size());
flt_ (u32, i, 0, (u32)v.size()) _1.push_back(u32(lower_bound(_, v[i]) - begin(_)));
return {_, _1};
}
} // namespace tifa_libs
#line 2 "src/util/rand/lib.hpp"
#line 5 "src/util/rand/lib.hpp"
namespace tifa_libs {
template <class T>
requires std::is_arithmetic_v<T>
class rand_gen {
using res_t = std::conditional_t<sizeof(T) <= 4, u32, u64>;
using res_wt = std::conditional_t<sizeof(T) <= 4, u64, u128>;
// clang-format off
struct mt19937_param { static CEXP u32 w = 32, n = 624, m = 397, r = 31, a = 0x9908b0df, u = 11, d = 0xffffffff, s = 7, b = 0x9d2c5680, t = 15, c = 0xefc60000, l = 18, f = 1812433253; };
struct mt19937_64_param { static CEXP u64 w = 64, n = 312, m = 156, r = 31, a = 0xb5026f5aa96619e9, u = 29, d = 0x5555555555555555, s = 17, b = 0x71d67fffeda60000, t = 37, c = 0xfff7eee000000000, l = 43, f = 6364136223846793005; };
using pm = std::conditional_t<std::is_same_v<res_t, u32>, mt19937_param, mt19937_64_param>;
// clang-format on
T a_, b_;
arr<res_t, pm::n> x_;
u32 p_;
CEXP void gen_() NE {
CEXP res_t um = (~res_t()) << pm::r, lm = ~um;
res_t _;
flt_ (res_t, i, p_ = 0, pm::n - pm::m) _ = ((x_[i] & um) | (x_[i + 1] & lm)), x_[i] = (x_[i + pm::m] ^ (_ >> 1) ^ ((_ & 1) ? pm::a : 0));
flt_ (res_t, i, pm::n - pm::m, pm::n - 1) _ = ((x_[i] & um) | (x_[i + 1] & lm)), x_[i] = (x_[i + (pm::m - pm::n)] ^ (_ >> 1) ^ ((_ & 1) ? pm::a : 0));
_ = ((x_[pm::n - 1] & um) | (x_[0] & lm)), x_[pm::n - 1] = (x_[pm::m - 1] ^ (_ >> 1) ^ ((_ & 1) ? pm::a : 0));
}
public:
CEXPE rand_gen(T a = std::numeric_limits<T>::min(), T b = std::numeric_limits<T>::max(), res_t sd = (res_t)TIME) NE : a_(a), b_(b) { assert(a < b || (std::is_integral_v<T> && a == b)), seed(sd); }
CEXP void range(T min, T max) NE { assert(min < max || (std::is_integral_v<T> && min == max)), a_ = min, b_ = max; }
void seed() NE { seed((res_t)std::chrono::duration_cast<std::chrono::nanoseconds>(std::chrono::high_resolution_clock::now().time_since_epoch()).count()); }
CEXP void seed(res_t sd) NE {
x_[0] = sd & gen_max();
flt_ (res_t, i, 1, p_ = pm::n) x_[i] = ((x_[i - 1] ^ (x_[i - 1] >> (pm::w - 2))) * pm::f + i % pm::n) & gen_max();
}
ND CEXP res_t gen_min() CNE { return 0; }
ND CEXP res_t gen_max() CNE {
if CEXP (sizeof(res_t) * 8 == pm::w) return ~res_t();
else return ((res_t)1 << pm::w) - 1;
}
CEXP res_t next() NE {
if (p_ >= pm::n) gen_();
res_t _ = x_[p_++];
_ ^= (_ >> pm::u) & pm::d, _ ^= (_ << pm::s) & pm::b, _ ^= (_ << pm::t) & pm::c, _ ^= (_ >> pm::l);
return _;
}
CEXP T operator()() NE {
if CEXP (std::integral<T>) {
const res_wt r = (res_wt)b_ - (res_wt)a_ + 1;
res_wt p = r * next();
if (auto l = (res_t)p, _ = res_t(res_wt(-(res_t)r) % r); l < r)
while (l < _) l = res_t(p = r * next());
return T((res_t)(p >> pm::w) + (res_t)a_);
} else return T(next() / (f128)((u128)gen_max() + 1) * (b_ - a_) + a_);
}
};
} // namespace tifa_libs
#line 2 "src/nt/gl/gcd/lib.hpp"
#line 4 "src/nt/gl/gcd/lib.hpp"
namespace tifa_libs {
namespace gcd_impl_ {
template <uint_c T, uint_c U>
CEXP std::common_type_t<T, U> gcd__(T u, U v) NE {
using W = std::common_type_t<T, U>;
retif_((!u || !v) [[unlikely]], u ^ v);
const auto k = std::__countr_zero(u | v);
u >>= k, v >>= k;
do {
if (W const _ = v >> std::__countr_zero(v); u > _) v = u - _, u = _;
else v = _ - u;
} while (v);
return u << k;
}
} // namespace gcd_impl_
template <int_c T, int_c U>
CEXP auto gcd(T a, U b) NE { return gcd_impl_::gcd__((to_uint_t<T>)abs(a), (to_uint_t<U>)abs(b)); }
} // namespace tifa_libs
#line 2 "src/nt/is_prime/lib.hpp"
#line 5 "src/nt/is_prime/lib.hpp"
namespace tifa_libs {
CEXP bool is_prime(u64 n) NE {
retif_((n <= 2) [[unlikely]], n == 2);
if (~n & 1) return false;
if (n < 8 || n == 61) return true;
if (!(n % 3) || !(n % 5) || !(n % 7)) return false;
if (n < 121) return true;
auto f = [n, d = (n - 1) >> std::countr_zero(n - 1)](auto&& bases) NE -> bool {
for (cu64 i : bases) {
if (!(i % n)) continue;
u64 t = d, y = qpow_mod(i, t, n);
while (t != n - 1 && y != 1 && y != n - 1) y = mul_mod_u(y, y, n), t *= 2;
if (y != n - 1 && (~t & 1)) return false;
}
return true;
};
// NOLINTBEGIN(modernize-avoid-c-arrays)
CEXP u64 THRES1 = 341531, BASE1[]{9345883071009581737u};
CEXP u64 THRES2 = 1050535501, BASE2[]{336781006125, 9639812373923155};
CEXP u64 THRES3 = 350269456337, BASE3[]{4230279247111683200, 14694767155120705706u, 16641139526367750375u};
CEXP u64 THRES4 = 55245642489451, BASE4[]{2, 141889084524735, 1199124725622454117, 11096072698276303650u};
CEXP u64 THRES5 = 7999252175582851, BASE5[]{2, 4130806001517, 149795463772692060, 186635894390467037, 3967304179347715805};
CEXP u64 THRES6 = 585226005592931977, BASE6[]{2, 123635709730000, 9233062284813009, 43835965440333360, 761179012939631437, 1263739024124850375};
CEXP u64 BASE7[]{2, 325, 9375, 28178, 450775, 9780504, 1795265022};
// NOLINTEND(modernize-avoid-c-arrays)
if (n < THRES1) return f(BASE1);
if (n < THRES2) return f(BASE2);
if (n < THRES3) return f(BASE3);
if (n < THRES4) return f(BASE4);
if (n < THRES5) return f(BASE5);
if (n < THRES6) return f(BASE6);
return f(BASE7);
}
} // namespace tifa_libs
#line 8 "src/nt/pfactors/lib.hpp"
namespace tifa_libs {
namespace pfactors_impl_ {
static rand_gen<u64> e;
static auto __ = [] { e.seed(); return 0; }();
CEXP u64 rho(u64 n) NE {
e.range(1, n - 1);
auto f = [n, r = e()](u64 x) NE { return (mul_mod_u(x, x, n) + r) % n; };
u64 g = 1, x = 0, y = e(), yy = 0;
cu32 LIM = 128;
for (u64 r = 1, q = 1; g == 1; r *= 2) {
x = y;
flt_ (u64, i, 0, r) y = f(y);
for (u64 k = 0; g == 1 && k < r; k += LIM) {
yy = y;
for (u64 i = 0; i < LIM && i < r - k; ++i) q = mul_mod_u(q, (n - (y = f(y)) + x) % n, n);
g = gcd(q, n);
}
}
if (g == n) do {
g = gcd((x + (n - (yy = f(yy)))) % n, n);
} while (g == 1);
retif_((g == n), rho(n), g);
}
CEXP void run(u64 n, vecuu& p) NE {
if (n < 2) return;
if (is_prime(n)) return p.push_back(n);
cu64 g = rho(n);
run(n / g, p), run(g, p);
}
} // namespace pfactors_impl_
template <bool unique = true>
CEXP vecuu pfactors(u64 n) NE {
vecuu p;
if (cu32 _ = (u32)std::countr_zero(n) & 63; _) {
n >>= _;
if CEXP (unique) p.push_back(2);
else p.assign(_, 2);
}
if (n < 1000'000)
for (u32 i = 3; i <= n; ++i) {
if (n % i) continue;
if CEXP (unique) p.push_back(i);
do {
if CEXP (n /= i; !unique) p.push_back(i);
} while (!(n % i));
}
if (n < 2) return p;
pfactors_impl_::run(n, p);
if CEXP (unique) return uniq(p);
tifa_libs::sort(p);
return p;
}
CEXP vecp<u64, u32> pf_exp(u64 n) NE {
auto p = pfactors<false>(n);
vecp<u64, u32> ans;
for (u64 lst = 0; cu64 i : p)
if (i != lst) ans.emplace_back(lst = i, 1);
else ++ans.back().second;
return ans;
}
} // namespace tifa_libs
#line 2 "src/nt/proot/is/lib.hpp"
#line 4 "src/nt/proot/is/lib.hpp"
namespace tifa_libs {
template <std::unsigned_integral T, class It>
CEXP bool is_proot(T g, T m, It pf_begin, It pf_end) NE {
retif_((!g) [[unlikely]], false);
for (; pf_begin != pf_end; ++pf_begin)
if (qpow_mod(g, (m - 1) / *pf_begin, m) == 1) return false;
return true;
}
} // namespace tifa_libs
#line 5 "src/nt/proot/uint/lib.hpp"
namespace tifa_libs {
CEXP u64 proot(u64 m) NE {
retif_((m == 2) [[unlikely]], 1);
retif_((m == 3 || m == 5) [[unlikely]], 2);
if (m == 104857601 || m == 167772161 || m == 469762049 || m == 998244353 || m == 1004535809) return 3;
if (m == 1012924417) return 5;
if (m == 754974721) return 11;
const auto pf = pfactors(m - 1);
for (u64 g = 2;; ++g)
if (is_proot(g, m, begin(pf), end(pf))) return g;
}
} // namespace tifa_libs
#line 5 "src/conv/trans/ntt/lib.hpp"
namespace tifa_libs {
template <class mint>
struct ntt {
using data_t = mint;
static_assert(is_prime(mint::mod()) && (mint::mod() & 3) == 1, "MOD must be prime with 4k+1");
static CEXP u64 R = std::countr_zero(mint::mod() - 1), max_size = 1_u64 << R;
static CEXP mint G = proot(mint::mod());
private:
static inline arr<mint, R + 1> root, iroot, inv2;
static inline arr<mint, R - 1> rate, irate;
u32 sz{};
public:
CEXPE ntt() NE {
if (inv2[0].val()) return;
root[R] = qpow(G, mint::mod() >> R), iroot[R] = root[R].inv();
for (u32 i = R - 1; ~i; --i) {
root[i] = root[i + 1] * root[i + 1];
iroot[i] = iroot[i + 1] * iroot[i + 1];
}
mint prod(1), iprod(1);
flt_ (u32, i, 0, R - 1) {
rate[i] = prod * root[i + 2];
irate[i] = iprod * iroot[i + 2];
prod *= iroot[i + 2], iprod *= root[i + 2];
}
mint i2 = mint::mod() / 2 + 1;
inv2[0] = 1;
flt_ (u32, i, 0, R) inv2[i + 1] = inv2[i] * i2;
}
ND CEXP u32 size() CNE { return sz; }
CEXP void bzr(u32 len = max_size) NE {
cu32 n = std::bit_ceil(len);
assert(n <= max_size), sz = n;
}
CEXP void dif(vec<mint>& f, u32 n = 0) CNE {
if (assert(size()); !n) n = size();
if (f.size() < n) f.resize(n);
assert(std::has_single_bit(n) && n <= size());
cu32 l = (u32)std::countr_zero(n);
flt_ (u32, i, 0, l) {
cu32 w = 1 << (l - 1 - i), b = 1 << i;
mint z = 1;
flt_ (u32, j, 0, b) {
cu32 o = j << (l - i);
flt_ (u32, k, 0, w) {
mint x = f[o + k], y = f[o + k + w] * z;
f[o + k] = x + y, f[o + k + w] = x - y;
}
z *= rate[(u32)std::countr_zero(~j)];
}
}
}
CEXP void dit(vec<mint>& f, u32 n = 0) CNE {
assert(size());
if (!n) n = size();
if (f.size() < n) f.resize(n);
assert(std::has_single_bit(n) && n <= size());
cu32 l = (u32)std::countr_zero(n);
for (u32 i = l - 1; ~i; --i) {
cu32 w = 1 << (l - 1 - i), b = 1 << i;
mint z = 1;
flt_ (u32, j, 0, b) {
cu32 o = j << (l - i);
flt_ (u32, k, 0, w) {
mint x = f[o + k], y = f[o + k + w];
f[o + k] = x + y, f[o + k + w] = (x - y) * z;
}
z *= irate[(u32)std::countr_zero(~j)];
}
}
flt_ (u32, i, 0, n) f[i] *= inv2[l];
}
};
} // namespace tifa_libs
#line 2 "src/conv/add/dft/lib.hpp"
#line 2 "src/conv/add/naive/lib.hpp"
#line 4 "src/conv/add/naive/lib.hpp"
namespace tifa_libs {
CEXP inline u32 CONV_NAIVE_THRESHOLD = 16;
template <class U, class T = U>
requires(sizeof(U) <= sizeof(T))
CEXP vec<T> conv_naive(vec<U> CR l, vec<U> CR r, u32 ans_size = 0) NE {
retif_((l.empty() || r.empty()) [[unlikely]], {});
if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
vec<T> ans(ans_size);
u32 n = (u32)l.size(), m = (u32)r.size();
auto &&l_ = n < m ? r : l, &&r_ = n < m ? l : r;
if (n < m) swap(n, m);
flt_ (u32, i, 0, n)
flt_ (u32, j, 0, min(m, ans_size - i)) ans[i + j] += (T)l_[i] * (T)r_[j];
return ans;
}
} // namespace tifa_libs
#line 5 "src/conv/add/dft/lib.hpp"
namespace tifa_libs {
template <dft_c DFT_t, std::same_as<TPN DFT_t::data_t> DFT_data_t>
CEXP vec<DFT_data_t> conv_dft(DFT_t& dft, vec<DFT_data_t> l, vec<DFT_data_t> r, u32 ans_size = 0) NE {
if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
if (min(l.size(), r.size()) < CONV_NAIVE_THRESHOLD) return conv_naive(l, r, ans_size);
dft.bzr(max({(u32)l.size(), (u32)r.size(), min(u32(l.size() + r.size() - 1), ans_size)}));
dft.dif(l), dft.dif(r);
flt_ (u32, i, 0, dft.size()) l[i] *= r[i];
dft.dit(l), l.resize(ans_size);
return l;
}
template <class DFT_t, class mint, class T = u64>
CEXP vec<mint> conv_dft_um(DFT_t& dft, vec<T> CR l, vec<T> CR r, u32 ans_size = 0) NE {
if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
vec<mint> l_, r_;
for (l_.reserve(l.size()); auto CR i : l) l_.push_back(i);
for (r_.reserve(r.size()); auto CR i : r) r_.push_back(i);
return conv_dft(dft, l_, r_, ans_size);
}
} // namespace tifa_libs
#line 2 "src/conv/add/naive_mod/lib.hpp"
#line 5 "src/conv/add/naive_mod/lib.hpp"
namespace tifa_libs {
CEXP inline u32 CONV_NAIVE_MOD_THRESHOLD = 16;
CEXP vecuu conv_naive_mod(spnuu l, spnuu r, u64 mod, u32 ans_size = 0) NE {
retif_((l.empty() || r.empty()) [[unlikely]], {});
if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
vecuu ans(ans_size);
u32 n = (u32)l.size(), m = (u32)r.size();
auto &&l_ = n < m ? r : l, &&r_ = n < m ? l : r;
if (n < m) swap(n, m);
flt_ (u32, i, 0, n)
flt_ (u32, j, 0, min(m, ans_size - i)) ans[i + j] += mul_mod_u(l_[i], r_[j], mod);
for (auto& i : ans) i %= mod;
return ans;
}
} // namespace tifa_libs
#line 7 "src/conv/add/ntt3/lib.hpp"
namespace tifa_libs {
// 167772161, 469762049, 754974721
template <class mint0, class mint1, class mint2>
CEXP vecuu conv_3ntt_u64(ntt<mint0>& ntt0, ntt<mint1>& ntt1, ntt<mint2>& ntt2, vecuu CR l, vecuu CR r, u64 mod, u32 ans_size = 0) NE {
if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
if (min(l.size(), r.size()) < CONV_NAIVE_MOD_THRESHOLD) return conv_naive_mod(l, r, mod, ans_size);
CEXP u64 m0 = mint0::mod(), m1 = mint1::mod(), m2 = mint2::mod();
cu64 r01 = mint1(m0).inv().val(), r02 = mint2(m0).inv().val(), r12 = mint2(m1).inv().val(),
r02r12 = (u32)mul_mod_u(r02, r12, m2),
w1 = m0 % mod, w2 = mul_mod_u(m0, m1, mod);
cvec<mint0> d0 = conv_dft_um<ntt<mint0>, mint0>(ntt0, l, r, ans_size);
cvec<mint1> d1 = conv_dft_um<ntt<mint1>, mint1>(ntt1, l, r, ans_size);
cvec<mint2> d2 = conv_dft_um<ntt<mint2>, mint2>(ntt2, l, r, ans_size);
vecuu ret(ans_size);
const barrett<0> brt_m1_r01(m1, r01), brt_m2_r02r12(m2, r02r12), brt_m2_r12(m2, r12), brt_mod_w1(mod, w1), brt_mod_w2(mod, w2);
flt_ (u32, i, 0, ans_size) {
cu64 n1 = d1[i].val(), n2 = d2[i].val(), a = d0[i].val(),
b = brt_m1_r01.reduce(n1 + m1 - a),
c = brt_m2_r02r12.reduce(n2 + m2 - a) + brt_m2_r12.reduce(m2 - b);
ret[i] = (a + brt_mod_w1.reduce(b) + brt_mod_w2.reduce(c % m2)) % mod;
}
return ret;
}
template <class mint, class mint0, class mint1, class mint2>
CEXP vec<mint> conv_3ntt(ntt<mint0>& ntt0, ntt<mint1>& ntt1, ntt<mint2>& ntt2, vec<mint> CR l, vec<mint> CR r, u32 ans_size = 0) NE {
if (!ans_size) ans_size = u32(l.size() + r.size() - 1);
if (min(l.size(), r.size()) < CONV_NAIVE_THRESHOLD) return conv_naive(l, r, ans_size);
vecuu l_(l.size()), r_(r.size());
flt_ (u32, i, 0, (u32)l.size()) l_[i] = l[i].val();
flt_ (u32, i, 0, (u32)r.size()) r_[i] = r[i].val();
vecuu _ = conv_3ntt_u64(ntt0, ntt1, ntt2, l_, r_, mint::mod(), ans_size);
vec<mint> res(_.size());
flt_ (u32, i, 0, (u32)_.size()) res[i] = _[i];
return res;
}
} // namespace tifa_libs
#line 2 "src/fps/ds/poly_c/lib.hpp"
#line 2 "src/util/strip/lib.hpp"
#line 4 "src/util/strip/lib.hpp"
namespace tifa_libs {
// pred(x) == true ==> drop
template <common_range R, class F>
CEXP auto lstrip_view(R CR range, F&& pred) NE {
auto v = range | views::drop_while(std::forward<F>(pred));
return subrange{begin(v), end(v)};
}
// pred(x) == true ==> drop
template <common_range R, class F>
CEXP auto rstrip_view(R CR range, F&& pred) NE {
auto v = range | views::reverse | views::drop_while(std::forward<F>(pred));
return subrange{end(v).base(), begin(v).base()};
}
// pred(x) == true ==> drop
template <common_range R, class F>
CEXP auto strip_view(R CR range, F&& pred) NE {
auto v = range | views::drop_while(std::forward<F>(pred)) | views::reverse | views::drop_while(std::forward<F>(pred));
return subrange{end(v).base(), begin(v).base()};
}
} // 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/fps/ds/poly_c/lib.hpp"
namespace tifa_libs {
// clang-format off
enum class CCORE : u8 { FFT_R2, NTT3, NTT };
// clang-format on
namespace poly_impl_ {
template <class ccore>
requires requires(ccore cc, vec<TPN ccore::val_t> l, vec<TPN ccore::val_t> r, u32 sz) {
{ ccore::ct_cat } -> std::same_as<CCORE CR>;
cc.conv(l, r), cc.conv(l, r, sz);
}
struct poly : vec<TPN ccore::val_t> {
using ccore_t = ccore;
using val_t = ccore_t::val_t;
using data_t = vec<val_t>;
static inline ccore_t conv_core;
CEXPE poly(u32 sz = 1, cT_(val_t) val = val_t{}) NE : data_t(sz, val) {}
CEXP poly(TPN data_t::const_iterator begin, TPN data_t::const_iterator end) NE : data_t(begin, end) {}
CEXP poly(data_t CR v) NE : data_t(v) {}
CEXP poly(data_t&& v) NE : data_t(std::move(v)) {}
CEXP poly(itl<val_t> v) NE : data_t(v) {}
CEXP poly(common_range auto CR v) NE : data_t(begin(v), end(v)) {}
friend CEXP auto& operator>>(istream_c auto& is, poly& poly) NE {
for (auto& val : poly) is >> val;
return is;
}
friend CEXP auto& operator<<(ostream_c auto& os, poly CR poly) NE {
retif_((!poly.size()) [[unlikely]], os);
flt_ (u32, i, 1, (u32)poly.size()) os << poly[i - 1] << ' ';
return os << poly.back();
}
ND CEXP bool is_zero() CNE {
for (auto&& i : *this)
if (i != 0) return false;
return true;
}
CEXP val_t operator()(val_t x) CNE {
val_t ans = 0;
for (u32 i = data_t::size() - 1; ~i; --i) ans = ans * x + data_t::data()[i];
return ans;
}
template <class F>
requires requires(F f, u32 idx, val_t& val) { f(idx, val); }
CEXP void apply_range(u32 l, u32 r, F&& f) NE {
assert(l < r && r <= data_t::size());
flt_ (u32, i, l, r) f(i, data_t::data()[i]);
}
template <class F>
CEXP void apply(F&& f) NE { apply_range(0, (u32)data_t::size(), std::forward<F>(f)); }
ND CEXP poly pre(u32 sz) CNE {
if (sz <= this->size()) return {this->begin(), this->begin() + sz};
poly _ = *this;
_.resize(sz);
return _;
}
CEXP void strip() NE {
auto [_, r] = rstrip_view(*this, [](cT_(val_t) x) NE { return x.val() == 0; });
if (data_t::erase(r, this->end()); data_t::empty()) data_t::push_back(val_t(0));
}
friend poly stripped(poly CR p) NE {
poly ret(rstrip_view(p, [](cT_(val_t) x) NE { return x.val() == 0; }));
if (ret.empty()) return {0};
return ret;
}
CEXP void reverse(u32 n = 0) NE { std::ranges::reverse(data_t::begin(), data_t::begin() + (n ? n : (u32)data_t::size())); }
CEXP void conv(poly CR r, u32 ans_size = 0) NE { conv_core.conv(*this, r, ans_size); }
CEXP poly operator-() CNE {
poly ret = *this;
ret.apply([](u32, auto& v) NE { v = -v; });
return ret;
}
friend CEXP poly operator+(poly p, val_t c) NE {
p[0] += c;
return p;
}
friend CEXP poly operator+(val_t c, poly CR p) NE { return p + c; }
friend CEXP poly operator-(poly p, val_t c) NE {
p[0] -= c;
return p;
}
friend CEXP poly operator-(val_t c, poly CR p) NE { return p - c; }
CEXP poly& operator*=(val_t c) NE {
apply([&c](u32, auto& v) NE { v *= c; });
return *this;
}
friend CEXP poly operator*(poly p, val_t c) NE { return p *= c; }
friend CEXP poly operator*(val_t c, poly p) NE { return p *= c; }
CEXP poly& operator+=(poly CR r) NE {
retif_((r.empty()) [[unlikely]], *this);
data_t::resize(max(data_t::size(), r.size())), apply_range(0, (u32)r.size(), [&r](u32 i, auto& v) NE { v += r[i]; });
return *this;
}
friend CEXP poly operator+(poly l, poly CR r) NE { return l += r; }
CEXP poly& operator-=(poly CR r) NE {
retif_((r.empty()) [[unlikely]], *this);
data_t::resize(max(data_t::size(), r.size()));
apply_range(0, (u32)r.size(), [&r](u32 i, auto& v) NE { v -= r[i]; });
return *this;
}
friend CEXP poly operator-(poly l, poly CR r) NE { return l -= r; }
CEXP poly& operator*=(poly CR r) NE {
if (r.empty()) {
data_t::resize(1), *data_t::data() = 0;
return *this;
}
conv(r);
return *this;
}
friend CEXP poly operator*(poly l, poly CR r) NE { return l *= r; }
CEXP auto operator<=>(poly CR r) CNE {
auto l_ = stripped(*this), r_ = stripped(r);
if (l_.size() != r_.size()) return l_.size() <=> r_.size();
return std::lexicographical_compare_three_way(l_.rbegin(), l_.rend(), r_.rbegin(), r_.rend());
}
CEXP bool operator==(poly CR r) CNE { return std::is_eq(*this <=> r); }
};
} // namespace poly_impl_
template <class T>
concept poly_c = std::same_as<T, poly_impl_::poly<TPN T::ccore_t>>;
} // namespace tifa_libs
#line 5 "src/fps/ds/ntt3/lib.hpp"
namespace tifa_libs {
namespace poly3ntt_impl_ {
template <class mint, class mint0, class mint1, class mint2>
struct cconv_3ntt {
using val_t = mint;
// clang-format off
struct ntt3 { ntt<mint0> _0; ntt<mint1> _1; ntt<mint2> _2; } ccore;
// clang-format on
static CEXP auto ct_cat = CCORE::NTT3;
CEXP void conv(vec<val_t>& l, vec<val_t> CR r, u32 sz = 0) NE { l = conv_3ntt<val_t, mint0, mint1, mint2>(ccore._0, ccore._1, ccore._2, l, r, sz); }
};
} // namespace poly3ntt_impl_
template <class mint, class mint0, class mint1, class mint2>
using poly3ntt = poly_impl_::poly<poly3ntt_impl_::cconv_3ntt<mint, mint0, mint1, mint2>>;
} // namespace tifa_libs
#line 2 "src/math/ds/mint/bs/lib.hpp"
#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 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/bs/lib.hpp"
namespace tifa_libs {
template <u64 MOD_>
class mint_bs_tag : public mint_impl_::mint_tag_base {
static_assert(MOD_ && MOD_ <= UINT32_MAX);
using core = barrett<MOD_>;
public:
static CEXP bool FIXED_MOD = true;
protected:
using raw_t = u32;
raw_t v_{};
CEXP mint_bs_tag() NE = default;
CEXP mint_bs_tag(int_c auto v) NE : v_{mod(v)} {}
public:
static CEXP raw_t mod(sint_c auto v) NE {
if (v >= 0) return mod((to_uint_t<decltype(v)>)v);
if (auto ret = mod((to_uint_t<decltype(v)>)-v); ret) return mod() - ret;
else return ret;
}
static CEXP raw_t mod(uint_c auto v) NE {
if CEXP (umost64_c<decltype(v)>) return (raw_t)core::reduce((u64)v);
else if (v < UINT64_MAX) return (raw_t)core::reduce((u64)v);
else return raw_t(v % mod());
}
static CEXP raw_t mod() NE { return MOD_; }
ND CEXP raw_t val() CNE { return v_; }
CEXP raw_t& data() NE { return v_; }
protected:
template <class mint>
CEXP auto neg() CNE {
mint res;
if (v_) res.v_ = mod() - v_;
return res;
}
CEXP void add(mint_bs_tag CR r) NE {
if ((v_ += r.v_) >= mod()) v_ -= mod();
}
CEXP void sub(mint_bs_tag CR r) NE {
if (i32(v_ -= r.v_) < 0) v_ += mod();
}
CEXP void mul(mint_bs_tag CR r) NE { v_ = (raw_t)core::reduce(u64(v_) * r.v_); }
};
template <u64 MOD>
using mint_bs = mint_impl_::mint<mint_bs_tag<MOD>>;
} // namespace tifa_libs
#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 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 2 "src/math/fact/helper_l/lib.hpp"
#line 2 "src/fps/ctsh/lib.hpp"
#line 5 "src/fps/ctsh/lib.hpp"
namespace tifa_libs {
template <poly_c poly_t, class T>
CEXP poly_t ctsh_fps(poly_t CR f, TPN poly_t::val_t c, vec<T> CR ifact, u32 m = 0) NE {
cu32 n = (u32)f.size(), k = n - 1;
using mint = TPN poly_t::val_t;
if (!m) m = n;
cu64 t = c.val();
if (t <= k) {
poly_t ret(m);
u32 ptr = 0;
for (u32 i = (u32)t; i <= k && ptr < m; ++i) ret[ptr++] = f[i];
if (k + 1 < t + m) {
auto suf = ctsh_fps(f, mint(k + 1), ifact, m - ptr);
flt_ (u64, i, k + 1, t + m) ret[ptr++] = suf[(u32)i - (k + 1)];
}
return ret;
}
if (t + m > mint::mod()) {
auto pre = ctsh_fps(f, mint(t), ifact, u32(mint::mod() - t)),
suf = ctsh_fps(f, mint(0), ifact, m - (u32)pre.size());
copy(suf, std::back_inserter(pre));
return pre;
}
poly_t d(k + 1);
flt_ (u32, i, 0, k + 1) {
d[i] = ifact[i], (d[i] *= ifact[k - i]) *= f[i];
if ((k - i) & 1) d[i] = -d[i];
}
poly_t h(m + k);
for (u32 i = 0; i < m + k; ++i) h[i] = mint(t - k + i).inv();
auto dh = d * h;
poly_t ret(m);
mint cur = t;
flt_ (u32, i, 1, k + 1) cur *= t - i;
flt_ (u32, i, 0, m) ret[i] = cur * dh[k + i], (cur *= t + i + 1) *= h[i];
return ret;
}
template <poly_c poly_t>
CEXP auto ctsh_fps(poly_t CR f, TPN poly_t::val_t c, u32 m = 0) NE { return ctsh_fps(f, c, gen_ifact((u32)f.size(), poly_t::val_t::mod()), m); }
} // namespace tifa_libs
#line 5 "src/math/fact/helper_l/lib.hpp"
namespace tifa_libs {
template <poly_c poly_t>
struct factl_helper : fact_helper<TPN poly_t::val_t> {
using val_t = TPN poly_t::val_t;
using base_t = fact_helper<TPN poly_t::val_t>;
static CEXP u32 LBSZ = 9;
static CEXP u32 SZ = 1 << LBSZ;
static CEXP u32 threshold = 2'000'001;
static_assert(threshold > SZ * 2);
factl_helper() = delete;
static CEXP void ensure(u32 sz = threshold) NE {
if (sz < threshold) [[unlikely]]
sz = threshold;
base_t::ensure(sz);
}
static CEXP val_t get_fact(u64 n) NE {
if (n >= base_t::mod()) [[unlikely]]
return 0;
if (base_t::fact.size() < threshold) [[unlikely]]
base_t::ensure(threshold);
retif_((n < base_t::fact.size()), base_t::fact[n], fact_(n));
}
static CEXP val_t get_ifact(u64 n) NE {
if (n >= base_t::mod()) [[unlikely]]
return 0;
if (base_t::fact.size() < threshold) [[unlikely]]
base_t::ensure(threshold);
retif_((n < base_t::fact.size()), base_t::ifact[n], fact_(n).inv());
}
private:
// mod() / SZ
static inline u64 B;
// f[i] = (i*B + 1) * ... * (i*B + B)
static inline poly_t f;
static val_t fact_(u64 n) NE {
static bool inited = false;
if (!inited) {
inited = true;
B = base_t::mod() >> LBSZ;
f.reserve(SZ), f[0] = 1;
flt_ (u32, i, 0, LBSZ) {
poly_t g = ctsh_fps(f, val_t(1 << i), base_t::ifact, 3_u32 << i);
const auto get = [&](u32 j) { retif_((j < (1_u32 << i)), f[j], g[j - (1 << i)]); };
f.resize(2_u32 << i);
flt_ (u32, j, 0, 2_u32 << i) f[j] = get(2 * j) * get(2 * j + 1) * ((2 * j + 1) << i);
}
if (B > SZ) {
vec<val_t> g = ctsh_fps(f, val_t(SZ), base_t::ifact, u32(B - SZ));
move(g, std::back_inserter(f));
} else f.resize(B);
flt_ (u32, i, 0, (u32)B) f[i] *= val_t(i + 1) * SZ;
f.insert(begin(f), 1);
flt_ (u32, i, 0, (u32)B) f[i + 1] *= f[i];
}
val_t res;
cu64 q = n / SZ;
cu32 r = n % SZ;
if (2 * r <= SZ) {
res = f[q];
flt_ (u32, i, 0, r) res *= n - i;
} else if (q != B) {
res = f[q + 1];
val_t den = 1;
flt_ (u32, i, 1, SZ - r + 1) den *= n + i;
res /= den;
} else {
res = base_t::mod() - 1;
val_t den = 1;
flt_ (u64, i, n + 1, base_t::mod()) den *= i;
res /= den;
}
return res;
}
};
} // namespace tifa_libs
#line 13 "test/cpv/library-checker-other/sum_of_exponential_times_polynomial.mintd-md64.mints-bs.poly_anymod-p3ntt.factorial-factl_helper.cpp"
using namespace tifa_libs;
using mint = mint_md64<__LINE__>;
using namespace tifa_libs;
using mint_p3ntt1 = mint_bs<167772161>;
using mint_p3ntt2 = mint_bs<469762049>;
using mint_p3ntt3 = mint_bs<754974721>;
using poly = poly3ntt<mint, mint_p3ntt1, mint_p3ntt2, mint_p3ntt3>;
using namespace tifa_libs;
using fact_t = factl_helper<poly>;
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.mints-bs.poly_anymod-p3ntt.factorial-factl_helper.cpp