#line 1 "/home/maspy/compro/library/my_template.hpp" #if defined(LOCAL) #include #else #pragma GCC optimize("Ofast") #pragma GCC optimize("unroll-loops") #include using namespace std; using ll = long long; using u32 = unsigned int; using u64 = unsigned long long; using i128 = __int128; using u128 = unsigned __int128; using f128 = __float128; template constexpr T infty = 0; template <> constexpr int infty = 1'000'000'000; template <> constexpr ll infty = ll(infty) * infty * 2; template <> constexpr u32 infty = infty; template <> constexpr u64 infty = infty; template <> constexpr i128 infty = i128(infty) * infty; template <> constexpr double infty = infty; template <> constexpr long double infty = infty; using pi = pair; using vi = vector; template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using vvvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define vv(type, name, h, ...) \ vector> name(h, vector(__VA_ARGS__)) #define vvv(type, name, h, w, ...) \ vector>> name( \ h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name( \ a, vector>>( \ b, vector>(c, vector(__VA_ARGS__)))) // https://trap.jp/post/1224/ #define FOR1(a) for (ll _ = 0; _ < ll(a); ++_) #define FOR2(i, a) for (ll i = 0; i < ll(a); ++i) #define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i) #define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c)) #define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i) #define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i) #define overload4(a, b, c, d, e, ...) e #define overload3(a, b, c, d, ...) d #define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__) #define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__) #define FOR_subset(t, s) \ for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s))) #define all(x) x.begin(), x.end() #define len(x) ll(x.size()) #define elif else if #define eb emplace_back #define mp make_pair #define mt make_tuple #define fi first #define se second #define stoi stoll int popcnt(int x) { return __builtin_popcount(x); } int popcnt(u32 x) { return __builtin_popcount(x); } int popcnt(ll x) { return __builtin_popcountll(x); } int popcnt(u64 x) { return __builtin_popcountll(x); } int popcnt_mod_2(int x) { return __builtin_parity(x); } int popcnt_mod_2(u32 x) { return __builtin_parity(x); } int popcnt_mod_2(ll x) { return __builtin_parityll(x); } int popcnt_mod_2(u64 x) { return __builtin_parityll(x); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2) int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); } int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); } // (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2) int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); } int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); } template T floor(T a, T b) { return a / b - (a % b && (a ^ b) < 0); } template T ceil(T x, T y) { return floor(x + y - 1, y); } template T bmod(T x, T y) { return x - y * floor(x, y); } template pair divmod(T x, T y) { T q = floor(x, y); return {q, x - q * y}; } template T SUM(const vector &A) { T sm = 0; for (auto &&a: A) sm += a; return sm; } #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define LB(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define UB(c, x) distance((c).begin(), upper_bound(all(c), (x))) #define UNIQUE(x) \ sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit() template T POP(deque &que) { T a = que.front(); que.pop_front(); return a; } template T POP(pq &que) { T a = que.top(); que.pop(); return a; } template T POP(pqg &que) { T a = que.top(); que.pop(); return a; } template T POP(vc &que) { T a = que.back(); que.pop_back(); return a; } template ll binary_search(F check, ll ok, ll ng, bool check_ok = true) { if (check_ok) assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; (check(x) ? ok : ng) = x; } return ok; } template double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; (check(x) ? ok : ng) = x; } return (ok + ng) / 2; } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } // ? は -1 vc s_to_vi(const string &S, char first_char) { vc A(S.size()); FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); } return A; } template vector cumsum(vector &A, int off = 1) { int N = A.size(); vector B(N + 1); FOR(i, N) { B[i + 1] = B[i] + A[i]; } if (off == 0) B.erase(B.begin()); return B; } // stable sort template vector argsort(const vector &A) { vector ids(len(A)); iota(all(ids), 0); sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); }); return ids; } // A[I[0]], A[I[1]], ... template vc rearrange(const vc &A, const vc &I) { vc B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } #endif #line 1 "/home/maspy/compro/library/other/io.hpp" #define FASTIO #include // https://judge.yosupo.jp/submission/21623 namespace fastio { static constexpr uint32_t SZ = 1 << 17; char ibuf[SZ]; char obuf[SZ]; char out[100]; // pointer of ibuf, obuf uint32_t pil = 0, pir = 0, por = 0; struct Pre { char num[10000][4]; constexpr Pre() : num() { for (int i = 0; i < 10000; i++) { int n = i; for (int j = 3; j >= 0; j--) { num[i][j] = n % 10 | '0'; n /= 10; } } } } constexpr pre; inline void load() { memcpy(ibuf, ibuf + pil, pir - pil); pir = pir - pil + fread(ibuf + pir - pil, 1, SZ - pir + pil, stdin); pil = 0; if (pir < SZ) ibuf[pir++] = '\n'; } inline void flush() { fwrite(obuf, 1, por, stdout); por = 0; } void rd(char &c) { do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); } void rd(string &x) { x.clear(); char c; do { if (pil + 1 > pir) load(); c = ibuf[pil++]; } while (isspace(c)); do { x += c; if (pil == pir) load(); c = ibuf[pil++]; } while (!isspace(c)); } template void rd_real(T &x) { string s; rd(s); x = stod(s); } template void rd_integer(T &x) { if (pil + 100 > pir) load(); char c; do c = ibuf[pil++]; while (c < '-'); bool minus = 0; if constexpr (is_signed::value || is_same_v) { if (c == '-') { minus = 1, c = ibuf[pil++]; } } x = 0; while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; } if constexpr (is_signed::value || is_same_v) { if (minus) x = -x; } } void rd(int &x) { rd_integer(x); } void rd(ll &x) { rd_integer(x); } void rd(i128 &x) { rd_integer(x); } void rd(u32 &x) { rd_integer(x); } void rd(u64 &x) { rd_integer(x); } void rd(u128 &x) { rd_integer(x); } void rd(double &x) { rd_real(x); } void rd(long double &x) { rd_real(x); } void rd(f128 &x) { rd_real(x); } template void rd(pair &p) { return rd(p.first), rd(p.second); } template void rd_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); rd(x); rd_tuple(t); } } template void rd(tuple &tpl) { rd_tuple(tpl); } template void rd(array &x) { for (auto &d: x) rd(d); } template void rd(vc &x) { for (auto &d: x) rd(d); } void read() {} template void read(H &h, T &... t) { rd(h), read(t...); } void wt(const char c) { if (por == SZ) flush(); obuf[por++] = c; } void wt(const string s) { for (char c: s) wt(c); } void wt(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) wt(s[i]); } template void wt_integer(T x) { if (por > SZ - 100) flush(); if (x < 0) { obuf[por++] = '-', x = -x; } int outi; for (outi = 96; x >= 10000; outi -= 4) { memcpy(out + outi, pre.num[x % 10000], 4); x /= 10000; } if (x >= 1000) { memcpy(obuf + por, pre.num[x], 4); por += 4; } else if (x >= 100) { memcpy(obuf + por, pre.num[x] + 1, 3); por += 3; } else if (x >= 10) { int q = (x * 103) >> 10; obuf[por] = q | '0'; obuf[por + 1] = (x - q * 10) | '0'; por += 2; } else obuf[por++] = x | '0'; memcpy(obuf + por, out + outi + 4, 96 - outi); por += 96 - outi; } template void wt_real(T x) { ostringstream oss; oss << fixed << setprecision(15) << double(x); string s = oss.str(); wt(s); } void wt(int x) { wt_integer(x); } void wt(ll x) { wt_integer(x); } void wt(i128 x) { wt_integer(x); } void wt(u32 x) { wt_integer(x); } void wt(u64 x) { wt_integer(x); } void wt(u128 x) { wt_integer(x); } void wt(double x) { wt_real(x); } void wt(long double x) { wt_real(x); } void wt(f128 x) { wt_real(x); } template void wt(const pair val) { wt(val.first); wt(' '); wt(val.second); } template void wt_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { wt(' '); } const auto x = std::get(t); wt(x); wt_tuple(t); } } template void wt(tuple tpl) { wt_tuple(tpl); } template void wt(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } template void wt(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } void print() { wt('\n'); } template void print(Head &&head, Tail &&... tail) { wt(head); if (sizeof...(Tail)) wt(' '); print(forward(tail)...); } // gcc expansion. called automaticall after main. void __attribute__((destructor)) _d() { flush(); } } // namespace fastio using fastio::read; using fastio::print; using fastio::flush; #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ read(__VA_ARGS__) #define U32(...) \ u32 __VA_ARGS__; \ read(__VA_ARGS__) #define U64(...) \ u64 __VA_ARGS__; \ read(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ read(__VA_ARGS__) #define CHAR(...) \ char __VA_ARGS__; \ read(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ read(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ read(name) #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ read(name) void YES(bool t = 1) { print(t ? "YES" : "NO"); } void NO(bool t = 1) { YES(!t); } void Yes(bool t = 1) { print(t ? "Yes" : "No"); } void No(bool t = 1) { Yes(!t); } void yes(bool t = 1) { print(t ? "yes" : "no"); } void no(bool t = 1) { yes(!t); } #line 3 "main.cpp" #line 2 "/home/maspy/compro/library/mod/modint_common.hpp" struct has_mod_impl { template static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{}); template static auto check(...) -> std::false_type; }; template class has_mod : public decltype(has_mod_impl::check(std::declval())) {}; template mint inv(int n) { static const int mod = mint::get_mod(); static vector dat = {0, 1}; assert(0 <= n); if (n >= mod) n %= mod; while (len(dat) <= n) { int k = len(dat); int q = (mod + k - 1) / k; dat.eb(dat[k * q - mod] * mint::raw(q)); } return dat[n]; } template mint fact(int n) { static const int mod = mint::get_mod(); assert(0 <= n && n < mod); static vector dat = {1, 1}; while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat))); return dat[n]; } template mint fact_inv(int n) { static vector dat = {1, 1}; if (n < 0) return mint(0); while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv(len(dat))); return dat[n]; } template mint fact_invs(Ts... xs) { return (mint(1) * ... * fact_inv(xs)); } template mint multinomial(Head &&head, Tail &&... tail) { return fact(head) * fact_invs(std::forward(tail)...); } template mint C_dense(int n, int k) { static vvc C; static int H = 0, W = 0; auto calc = [&](int i, int j) -> mint { if (i == 0) return (j == 0 ? mint(1) : mint(0)); return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0); }; if (W <= k) { FOR(i, H) { C[i].resize(k + 1); FOR(j, W, k + 1) { C[i][j] = calc(i, j); } } W = k + 1; } if (H <= n) { C.resize(n + 1); FOR(i, H, n + 1) { C[i].resize(W); FOR(j, W) { C[i][j] = calc(i, j); } } H = n + 1; } return C[n][k]; } template mint C(ll n, ll k) { assert(n >= 0); if (k < 0 || n < k) return 0; if constexpr (dense) return C_dense(n, k); if constexpr (!large) return multinomial(n, k, n - k); k = min(k, n - k); mint x(1); FOR(i, k) x *= mint(n - i); return x * fact_inv(k); } template mint C_inv(ll n, ll k) { assert(n >= 0); assert(0 <= k && k <= n); if (!large) return fact_inv(n) * fact(k) * fact(n - k); return mint(1) / C(n, k); } // [x^d](1-x)^{-n} template mint C_negative(ll n, ll d) { assert(n >= 0); if (d < 0) return mint(0); if (n == 0) { return (d == 0 ? mint(1) : mint(0)); } return C(n + d - 1, d); } #line 3 "/home/maspy/compro/library/mod/modint.hpp" template struct modint { static constexpr u32 umod = u32(mod); static_assert(umod < u32(1) << 31); u32 val; static modint raw(u32 v) { modint x; x.val = v; return x; } constexpr modint() : val(0) {} constexpr modint(u32 x) : val(x % umod) {} constexpr modint(u64 x) : val(x % umod) {} constexpr modint(u128 x) : val(x % umod) {} constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){}; constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){}; constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){}; bool operator<(const modint &other) const { return val < other.val; } modint &operator+=(const modint &p) { if ((val += p.val) >= umod) val -= umod; return *this; } modint &operator-=(const modint &p) { if ((val += umod - p.val) >= umod) val -= umod; return *this; } modint &operator*=(const modint &p) { val = u64(val) * p.val % umod; return *this; } modint &operator/=(const modint &p) { *this *= p.inverse(); return *this; } modint operator-() const { return modint::raw(val ? mod - val : u32(0)); } modint operator+(const modint &p) const { return modint(*this) += p; } modint operator-(const modint &p) const { return modint(*this) -= p; } modint operator*(const modint &p) const { return modint(*this) *= p; } modint operator/(const modint &p) const { return modint(*this) /= p; } bool operator==(const modint &p) const { return val == p.val; } bool operator!=(const modint &p) const { return val != p.val; } modint inverse() const { int a = val, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } return modint(u); } modint pow(ll n) const { assert(n >= 0); modint ret(1), mul(val); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } static constexpr int get_mod() { return mod; } // (n, r), r は 1 の 2^n 乗根 static constexpr pair ntt_info() { if (mod == 120586241) return {20, 74066978}; if (mod == 167772161) return {25, 17}; if (mod == 469762049) return {26, 30}; if (mod == 754974721) return {24, 362}; if (mod == 880803841) return {23, 211}; if (mod == 943718401) return {22, 663003469}; if (mod == 998244353) return {23, 31}; if (mod == 1045430273) return {20, 363}; if (mod == 1051721729) return {20, 330}; if (mod == 1053818881) return {20, 2789}; return {-1, -1}; } static constexpr bool can_ntt() { return ntt_info().fi != -1; } }; #ifdef FASTIO template void rd(modint &x) { fastio::rd(x.val); x.val %= mod; // assert(0 <= x.val && x.val < mod); } template void wt(modint x) { fastio::wt(x.val); } #endif using modint107 = modint<1000000007>; using modint998 = modint<998244353>; #line 2 "/home/maspy/compro/library/poly/count_terms.hpp" template int count_terms(const vc& f){ int t = 0; FOR(i, len(f)) if(f[i] != mint(0)) ++t; return t; } #line 2 "/home/maspy/compro/library/mod/mod_inv.hpp" // long でも大丈夫 // (val * x - 1) が mod の倍数になるようにする // 特に mod=0 なら x=0 が満たす ll mod_inv(ll val, ll mod) { if (mod == 0) return 0; mod = abs(mod); val %= mod; if (val < 0) val += mod; ll a = val, b = mod, u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } if (u < 0) u += mod; return u; } #line 1 "/home/maspy/compro/library/mod/crt3.hpp" constexpr u32 mod_pow_constexpr(u64 a, u64 n, u32 mod) { a %= mod; u64 res = 1; FOR(32) { if (n & 1) res = res * a % mod; a = a * a % mod, n /= 2; } return res; } template T CRT3(u64 a0, u64 a1, u64 a2) { static_assert(p0 < p1 && p1 < p2); static constexpr u64 x0_1 = mod_pow_constexpr(p0, p1 - 2, p1); static constexpr u64 x01_2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2); u64 c = (a1 - a0 + p1) * x0_1 % p1; u64 a = a0 + c * p0; c = (a2 - a % p2 + p2) * x01_2 % p2; return T(a) + T(c) * T(p0) * T(p1); } #line 2 "/home/maspy/compro/library/poly/convolution_naive.hpp" template ::value>::type* = nullptr> vc convolution_naive(const vc& a, const vc& b) { int n = int(a.size()), m = int(b.size()); if (n > m) return convolution_naive(b, a); if (n == 0) return {}; vector ans(n + m - 1); FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j]; return ans; } template ::value>::type* = nullptr> vc convolution_naive(const vc& a, const vc& b) { int n = int(a.size()), m = int(b.size()); if (n > m) return convolution_naive(b, a); if (n == 0) return {}; vc ans(n + m - 1); if (n <= 16 && (T::get_mod() < (1 << 30))) { for (int k = 0; k < n + m - 1; ++k) { int s = max(0, k - m + 1); int t = min(n, k + 1); u64 sm = 0; for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); } ans[k] = sm; } } else { for (int k = 0; k < n + m - 1; ++k) { int s = max(0, k - m + 1); int t = min(n, k + 1); u128 sm = 0; for (int i = s; i < t; ++i) { sm += u64(a[i].val) * (b[k - i].val); } ans[k] = T::raw(sm % T::get_mod()); } } return ans; } #line 2 "/home/maspy/compro/library/poly/convolution_karatsuba.hpp" // 任意の環でできる template vc convolution_karatsuba(const vc& f, const vc& g) { const int thresh = 30; if (min(len(f), len(g)) <= thresh) return convolution_naive(f, g); int n = max(len(f), len(g)); int m = ceil(n, 2); vc f1, f2, g1, g2; if (len(f) < m) f1 = f; if (len(f) >= m) f1 = {f.begin(), f.begin() + m}; if (len(f) >= m) f2 = {f.begin() + m, f.end()}; if (len(g) < m) g1 = g; if (len(g) >= m) g1 = {g.begin(), g.begin() + m}; if (len(g) >= m) g2 = {g.begin() + m, g.end()}; vc a = convolution_karatsuba(f1, g1); vc b = convolution_karatsuba(f2, g2); FOR(i, len(f2)) f1[i] += f2[i]; FOR(i, len(g2)) g1[i] += g2[i]; vc c = convolution_karatsuba(f1, g1); vc F(len(f) + len(g) - 1); assert(2 * m + len(b) <= len(F)); FOR(i, len(a)) F[i] += a[i], c[i] -= a[i]; FOR(i, len(b)) F[2 * m + i] += b[i], c[i] -= b[i]; if (c.back() == T(0)) c.pop_back(); FOR(i, len(c)) if (c[i] != T(0)) F[m + i] += c[i]; return F; } #line 2 "/home/maspy/compro/library/poly/ntt.hpp" template void ntt(vector& a, bool inverse) { assert(mint::can_ntt()); const int rank2 = mint::ntt_info().fi; const int mod = mint::get_mod(); static array root, iroot; static array rate2, irate2; static array rate3, irate3; static bool prepared = 0; if (!prepared) { prepared = 1; root[rank2] = mint::ntt_info().se; iroot[rank2] = mint(1) / root[rank2]; FOR_R(i, rank2) { root[i] = root[i + 1] * root[i + 1]; iroot[i] = iroot[i + 1] * iroot[i + 1]; } mint prod = 1, iprod = 1; for (int i = 0; i <= rank2 - 2; i++) { rate2[i] = root[i + 2] * prod; irate2[i] = iroot[i + 2] * iprod; prod *= iroot[i + 2]; iprod *= root[i + 2]; } prod = 1, iprod = 1; for (int i = 0; i <= rank2 - 3; i++) { rate3[i] = root[i + 3] * prod; irate3[i] = iroot[i + 3] * iprod; prod *= iroot[i + 3]; iprod *= root[i + 3]; } } int n = int(a.size()); int h = topbit(n); assert(n == 1 << h); if (!inverse) { int len = 0; while (len < h) { if (h - len == 1) { int p = 1 << (h - len - 1); mint rot = 1; FOR(s, 1 << len) { int offset = s << (h - len); FOR(i, p) { auto l = a[i + offset]; auto r = a[i + offset + p] * rot; a[i + offset] = l + r; a[i + offset + p] = l - r; } rot *= rate2[topbit(~s & -~s)]; } len++; } else { int p = 1 << (h - len - 2); mint rot = 1, imag = root[2]; for (int s = 0; s < (1 << len); s++) { mint rot2 = rot * rot; mint rot3 = rot2 * rot; int offset = s << (h - len); for (int i = 0; i < p; i++) { u64 mod2 = u64(mod) * mod; u64 a0 = a[i + offset].val; u64 a1 = u64(a[i + offset + p].val) * rot.val; u64 a2 = u64(a[i + offset + 2 * p].val) * rot2.val; u64 a3 = u64(a[i + offset + 3 * p].val) * rot3.val; u64 a1na3imag = (a1 + mod2 - a3) % mod * imag.val; u64 na2 = mod2 - a2; a[i + offset] = a0 + a2 + a1 + a3; a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3)); a[i + offset + 2 * p] = a0 + na2 + a1na3imag; a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag); } rot *= rate3[topbit(~s & -~s)]; } len += 2; } } } else { mint coef = mint(1) / mint(len(a)); FOR(i, len(a)) a[i] *= coef; int len = h; while (len) { if (len == 1) { int p = 1 << (h - len); mint irot = 1; FOR(s, 1 << (len - 1)) { int offset = s << (h - len + 1); FOR(i, p) { u64 l = a[i + offset].val; u64 r = a[i + offset + p].val; a[i + offset] = l + r; a[i + offset + p] = (mod + l - r) * irot.val; } irot *= irate2[topbit(~s & -~s)]; } len--; } else { int p = 1 << (h - len); mint irot = 1, iimag = iroot[2]; FOR(s, (1 << (len - 2))) { mint irot2 = irot * irot; mint irot3 = irot2 * irot; int offset = s << (h - len + 2); for (int i = 0; i < p; i++) { u64 a0 = a[i + offset + 0 * p].val; u64 a1 = a[i + offset + 1 * p].val; u64 a2 = a[i + offset + 2 * p].val; u64 a3 = a[i + offset + 3 * p].val; u64 x = (mod + a2 - a3) * iimag.val % mod; a[i + offset] = a0 + a1 + a2 + a3; a[i + offset + 1 * p] = (a0 + mod - a1 + x) * irot.val; a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * irot2.val; a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * irot3.val; } irot *= irate3[topbit(~s & -~s)]; } len -= 2; } } } } #line 1 "/home/maspy/compro/library/poly/fft.hpp" namespace CFFT { using real = double; struct C { real x, y; C() : x(0), y(0) {} C(real x, real y) : x(x), y(y) {} inline C operator+(const C& c) const { return C(x + c.x, y + c.y); } inline C operator-(const C& c) const { return C(x - c.x, y - c.y); } inline C operator*(const C& c) const { return C(x * c.x - y * c.y, x * c.y + y * c.x); } inline C conj() const { return C(x, -y); } }; const real PI = acosl(-1); int base = 1; vector rts = {{0, 0}, {1, 0}}; vector rev = {0, 1}; void ensure_base(int nbase) { if (nbase <= base) return; rev.resize(1 << nbase); rts.resize(1 << nbase); for (int i = 0; i < (1 << nbase); i++) { rev[i] = (rev[i >> 1] >> 1) + ((i & 1) << (nbase - 1)); } while (base < nbase) { real angle = PI * 2.0 / (1 << (base + 1)); for (int i = 1 << (base - 1); i < (1 << base); i++) { rts[i << 1] = rts[i]; real angle_i = angle * (2 * i + 1 - (1 << base)); rts[(i << 1) + 1] = C(cos(angle_i), sin(angle_i)); } ++base; } } void fft(vector& a, int n) { assert((n & (n - 1)) == 0); int zeros = __builtin_ctz(n); ensure_base(zeros); int shift = base - zeros; for (int i = 0; i < n; i++) { if (i < (rev[i] >> shift)) { swap(a[i], a[rev[i] >> shift]); } } for (int k = 1; k < n; k <<= 1) { for (int i = 0; i < n; i += 2 * k) { for (int j = 0; j < k; j++) { C z = a[i + j + k] * rts[j + k]; a[i + j + k] = a[i + j] - z; a[i + j] = a[i + j] + z; } } } } } // namespace CFFT #line 9 "/home/maspy/compro/library/poly/convolution.hpp" template vector convolution_ntt(vector a, vector b) { if (a.empty() || b.empty()) return {}; int n = int(a.size()), m = int(b.size()); int sz = 1; while (sz < n + m - 1) sz *= 2; // sz = 2^k のときの高速化。分割統治的なやつで損しまくるので。 if ((n + m - 3) <= sz / 2) { auto a_last = a.back(), b_last = b.back(); a.pop_back(), b.pop_back(); auto c = convolution(a, b); c.resize(n + m - 1); c[n + m - 2] = a_last * b_last; FOR(i, len(a)) c[i + len(b)] += a[i] * b_last; FOR(i, len(b)) c[i + len(a)] += b[i] * a_last; return c; } a.resize(sz), b.resize(sz); bool same = a == b; ntt(a, 0); if (same) { b = a; } else { ntt(b, 0); } FOR(i, sz) a[i] *= b[i]; ntt(a, 1); a.resize(n + m - 1); return a; } template vector convolution_garner(const vector& a, const vector& b) { int n = len(a), m = len(b); if (!n || !m) return {}; static constexpr int p0 = 167772161; static constexpr int p1 = 469762049; static constexpr int p2 = 754974721; using mint0 = modint; using mint1 = modint; using mint2 = modint; vc a0(n), b0(m); vc a1(n), b1(m); vc a2(n), b2(m); FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val; FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val; auto c0 = convolution_ntt(a0, b0); auto c1 = convolution_ntt(a1, b1); auto c2 = convolution_ntt(a2, b2); vc c(len(c0)); FOR(i, n + m - 1) { c[i] = CRT3(c0[i].val, c1[i].val, c2[i].val); } return c; } template vc convolution_fft(const vc& a, const vc& b) { using C = CFFT::C; int need = (int)a.size() + (int)b.size() - 1; int nbase = 1; while ((1 << nbase) < need) nbase++; CFFT::ensure_base(nbase); int sz = 1 << nbase; vector fa(sz); for (int i = 0; i < sz; i++) { int x = (i < (int)a.size() ? a[i] : 0); int y = (i < (int)b.size() ? b[i] : 0); fa[i] = C(x, y); } CFFT::fft(fa, sz); C r(0, -0.25 / (sz >> 1)), s(0, 1), t(0.5, 0); for (int i = 0; i <= (sz >> 1); i++) { int j = (sz - i) & (sz - 1); C z = (fa[j] * fa[j] - (fa[i] * fa[i]).conj()) * r; fa[j] = (fa[i] * fa[i] - (fa[j] * fa[j]).conj()) * r; fa[i] = z; } for (int i = 0; i < (sz >> 1); i++) { C A0 = (fa[i] + fa[i + (sz >> 1)]) * t; C A1 = (fa[i] - fa[i + (sz >> 1)]) * t * CFFT::rts[(sz >> 1) + i]; fa[i] = A0 + A1 * s; } CFFT::fft(fa, sz >> 1); vector ret(need); for (int i = 0; i < need; i++) { ret[i] = (i & 1 ? fa[i >> 1].y : fa[i >> 1].x); } return ret; } vector convolution(const vector& a, const vector& b) { int n = len(a), m = len(b); if (!n || !m) return {}; if (min(n, m) <= 2500) return convolution_naive(a, b); ll abs_sum_a = 0, abs_sum_b = 0; ll LIM = 1e15; FOR(i, n) abs_sum_a = min(LIM, abs_sum_a + abs(a[i])); FOR(i, m) abs_sum_b = min(LIM, abs_sum_b + abs(b[i])); if (i128(abs_sum_a) * abs_sum_b < 1e15) { vc c = convolution_fft(a, b); vc res(len(c)); FOR(i, len(c)) res[i] = ll(floor(c[i] + .5)); return res; } static constexpr unsigned long long MOD1 = 754974721; // 2^24 static constexpr unsigned long long MOD2 = 167772161; // 2^25 static constexpr unsigned long long MOD3 = 469762049; // 2^26 static constexpr unsigned long long M2M3 = MOD2 * MOD3; static constexpr unsigned long long M1M3 = MOD1 * MOD3; static constexpr unsigned long long M1M2 = MOD1 * MOD2; static constexpr unsigned long long M1M2M3 = MOD1 * MOD2 * MOD3; static const unsigned long long i1 = mod_inv(MOD2 * MOD3, MOD1); static const unsigned long long i2 = mod_inv(MOD1 * MOD3, MOD2); static const unsigned long long i3 = mod_inv(MOD1 * MOD2, MOD3); using mint1 = modint; using mint2 = modint; using mint3 = modint; vc a1(n), b1(m); vc a2(n), b2(m); vc a3(n), b3(m); FOR(i, n) a1[i] = a[i], a2[i] = a[i], a3[i] = a[i]; FOR(i, m) b1[i] = b[i], b2[i] = b[i], b3[i] = b[i]; auto c1 = convolution_ntt(a1, b1); auto c2 = convolution_ntt(a2, b2); auto c3 = convolution_ntt(a3, b3); vc c(n + m - 1); FOR(i, n + m - 1) { u64 x = 0; x += (c1[i].val * i1) % MOD1 * M2M3; x += (c2[i].val * i2) % MOD2 * M1M3; x += (c3[i].val * i3) % MOD3 * M1M2; ll diff = c1[i].val - ((long long)(x) % (long long)(MOD1)); if (diff < 0) diff += MOD1; static constexpr unsigned long long offset[5] = {0, 0, M1M2M3, 2 * M1M2M3, 3 * M1M2M3}; x -= offset[diff % 5]; c[i] = x; } return c; } template vc convolution(const vc& a, const vc& b) { int n = len(a), m = len(b); if (!n || !m) return {}; if (mint::can_ntt()) { if (min(n, m) <= 50) return convolution_karatsuba(a, b); return convolution_ntt(a, b); } if (min(n, m) <= 200) return convolution_karatsuba(a, b); return convolution_garner(a, b); } #line 2 "/home/maspy/compro/library/poly/integrate.hpp" // 不定積分:integrate(f) // 定積分:integrate(f, L, R) template vc integrate(const vc& f) { vc g(len(f) + 1); FOR3(i, 1, len(g)) g[i] = f[i - 1] * inv(i); return g; } // 不定積分:integrate(f) // 定積分:integrate(f, L, R) template mint integrate(const vc& f, mint L, mint R) { mint I = 0; mint pow_L = 1, pow_R = 1; FOR(i, len(f)) { pow_L *= L, pow_R *= R; I += inv(i + 1) * f[i] * (pow_R - pow_L); } return I; } #line 2 "/home/maspy/compro/library/poly/differentiate.hpp" template vc differentiate(const vc& f) { if (len(f) <= 1) return {}; vc g(len(f) - 1); FOR(i, len(g)) g[i] = f[i + 1] * mint(i + 1); return g; } #line 6 "/home/maspy/compro/library/poly/fps_exp.hpp" template vc fps_exp_sparse(vc& f) { if (len(f) == 0) return {mint(1)}; assert(f[0] == 0); int N = len(f); // df を持たせる vc> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i - 1, mint(i) * f[i]); vc F(N); F[0] = 1; FOR(n, 1, N) { mint rhs = 0; for (auto&& [k, fk]: dat) { if (k > n - 1) break; rhs += fk * F[n - 1 - k]; } F[n] = rhs * inv(n); } return F; } template vc fps_exp_dense(vc& h) { const int n = len(h); assert(n > 0 && h[0] == mint(0)); if (mint::can_ntt()) { vc& f = h; vc b = {1, (1 < n ? f[1] : 0)}; vc c = {1}, z1, z2 = {1, 1}; while (len(b) < n) { int m = len(b); auto y = b; y.resize(2 * m); ntt(y, 0); z1 = z2; vc z(m); FOR(i, m) z[i] = y[i] * z1[i]; ntt(z, 1); FOR(i, m / 2) z[i] = 0; ntt(z, 0); FOR(i, m) z[i] *= -z1[i]; ntt(z, 1); c.insert(c.end(), z.begin() + m / 2, z.end()); z2 = c; z2.resize(2 * m); ntt(z2, 0); vc x(f.begin(), f.begin() + m); FOR(i, len(x) - 1) x[i] = x[i + 1] * mint(i + 1); x.back() = 0; ntt(x, 0); FOR(i, m) x[i] *= y[i]; ntt(x, 1); FOR(i, m - 1) x[i] -= b[i + 1] * mint(i + 1); x.resize(m + m); FOR(i, m - 1) x[m + i] = x[i], x[i] = 0; ntt(x, 0); FOR(i, m + m) x[i] *= z2[i]; ntt(x, 1); FOR_R(i, len(x) - 1) x[i + 1] = x[i] * inv(i + 1); x[0] = 0; FOR3(i, m, min(n, m + m)) x[i] += f[i]; FOR(i, m) x[i] = 0; ntt(x, 0); FOR(i, m + m) x[i] *= y[i]; ntt(x, 1); b.insert(b.end(), x.begin() + m, x.end()); } b.resize(n); return b; } const int L = len(h); assert(L > 0 && h[0] == mint(0)); int LOG = 0; while (1 << LOG < L) ++LOG; h.resize(1 << LOG); auto dh = differentiate(h); vc f = {1}, g = {1}; int m = 1; vc p; FOR(LOG) { p = convolution(f, g); p.resize(m); p = convolution(p, g); p.resize(m); g.resize(m); FOR(i, m) g[i] += g[i] - p[i]; p = {dh.begin(), dh.begin() + m - 1}; p = convolution(f, p); p.resize(m + m - 1); FOR(i, m + m - 1) p[i] = -p[i]; FOR(i, m - 1) p[i] += mint(i + 1) * f[i + 1]; p = convolution(p, g); p.resize(m + m - 1); FOR(i, m - 1) p[i] += dh[i]; p = integrate(p); FOR(i, m + m) p[i] = h[i] - p[i]; p[0] += mint(1); f = convolution(f, p); f.resize(m + m); m += m; } f.resize(L); return f; } template vc fps_exp(vc& f) { int n = count_terms(f); int t = (mint::can_ntt() ? 320 : 3000); return (n <= t ? fps_exp_sparse(f) : fps_exp_dense(f)); } #line 2 "/home/maspy/compro/library/poly/fps_log.hpp" #line 4 "/home/maspy/compro/library/poly/fps_inv.hpp" template vc fps_inv_sparse(const vc& f) { int N = len(f); vc> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i, f[i]); vc g(N); mint g0 = mint(1) / f[0]; g[0] = g0; FOR(n, 1, N) { mint rhs = 0; for (auto&& [k, fk]: dat) { if (k > n) break; rhs -= fk * g[n - k]; } g[n] = rhs * g0; } return g; } template vc fps_inv_dense_ntt(const vc& F) { vc G = {mint(1) / F[0]}; ll N = len(F), n = 1; G.reserve(N); while (n < N) { vc f(2 * n), g(2 * n); FOR(i, min(N, 2 * n)) f[i] = F[i]; FOR(i, n) g[i] = G[i]; ntt(f, false), ntt(g, false); FOR(i, 2 * n) f[i] *= g[i]; ntt(f, true); FOR(i, n) f[i] = 0; ntt(f, false); FOR(i, 2 * n) f[i] *= g[i]; ntt(f, true); FOR(i, n, min(N, 2 * n)) G.eb(-f[i]); n *= 2; } return G; } template vc fps_inv_dense(const vc& F) { if (mint::can_ntt()) return fps_inv_dense_ntt(F); const int N = len(F); vc R = {mint(1) / F[0]}; vc p; int m = 1; while (m < N) { p = convolution(R, R); p.resize(m + m); vc f = {F.begin(), F.begin() + min(m + m, N)}; p = convolution(p, f); R.resize(m + m); FOR(i, m + m) R[i] = R[i] + R[i] - p[i]; m += m; } R.resize(N); return R; } template vc fps_inv(const vc& f) { assert(f[0] != mint(0)); int n = count_terms(f); int t = (mint::can_ntt() ? 160 : 820); return (n <= t ? fps_inv_sparse(f) : fps_inv_dense(f)); } #line 5 "/home/maspy/compro/library/poly/fps_log.hpp" template vc fps_log_dense(const vc& f) { assert(f[0] == mint(1)); ll N = len(f); vc df = f; FOR(i, N) df[i] *= mint(i); df.erase(df.begin()); auto f_inv = fps_inv(f); auto g = convolution(df, f_inv); g.resize(N - 1); g.insert(g.begin(), 0); FOR(i, N) g[i] *= inv(i); return g; } template vc fps_log_sparse(const vc& f) { int N = f.size(); vc> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i, f[i]); vc F(N); vc g(N - 1); for (int n = 0; n < N - 1; ++n) { mint rhs = mint(n + 1) * f[n + 1]; for (auto&& [i, fi]: dat) { if (i > n) break; rhs -= fi * g[n - i]; } g[n] = rhs; F[n + 1] = rhs * inv(n + 1); } return F; } template vc fps_log(const vc& f) { assert(f[0] == mint(1)); int n = count_terms(f); int t = (mint::can_ntt() ? 200 : 1200); return (n <= t ? fps_log_sparse(f) : fps_log_dense(f)); } #line 5 "/home/maspy/compro/library/poly/fps_pow.hpp" // fps の k 乗を求める。k >= 0 の前提である。 // 定数項が 1 で、k が mint の場合には、fps_pow_1 を使うこと。 // ・dense な場合: log, exp を使う O(NlogN) // ・sparse な場合: O(NK) template vc fps_pow(const vc& f, ll k) { assert(0 <= k); int n = len(f); if (k == 0) { vc g(n); g[0] = mint(1); return g; } int d = n; FOR_R(i, n) if (f[i] != 0) d = i; // d * k >= n if (d >= ceil(n, k)) { vc g(n); return g; } ll off = d * k; mint c = f[d]; mint c_inv = mint(1) / mint(c); vc g(n - off); FOR(i, n - off) g[i] = f[d + i] * c_inv; g = fps_pow_1(g, mint(k)); vc h(n); c = c.pow(k); FOR(i, len(g)) h[off + i] = g[i] * c; return h; } template vc fps_pow_1_sparse(const vc& f, mint K) { int N = len(f); assert(N == 0 || f[0] == mint(1)); vc> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i, f[i]); vc g(N); g[0] = 1; FOR(n, N - 1) { mint& x = g[n + 1]; for (auto&& [d, cf]: dat) { if (d > n + 1) break; mint t = cf * g[n - d + 1]; x += t * (K * mint(d) - mint(n - d + 1)); } x *= inv(n + 1); } return g; } template vc fps_pow_1_dense(const vc& f, mint K) { assert(f[0] == mint(1)); auto log_f = fps_log(f); FOR(i, len(f)) log_f[i] *= K; return fps_exp_dense(log_f); } template vc fps_pow_1(const vc& f, mint K) { int n = count_terms(f); int t = (mint::can_ntt() ? 100 : 1300); return (n <= t ? fps_pow_1_sparse(f, K) : fps_pow_1_dense(f, K)); } // f^e, sparse, O(NMK) template vvc fps_pow_1_sparse_2d(vvc f, mint n) { assert(f[0][0] == mint(1)); int N = len(f), M = len(f[0]); vv(mint, dp, N, M); dp[0] = fps_pow_1_sparse(f[0], n); vc> dat; FOR(i, N) FOR(j, M) { if ((i > 0 || j > 0) && f[i][j] != mint(0)) dat.eb(i, j, f[i][j]); } FOR(i, 1, N) { FOR(j, M) { // F = f^n, f dF = n df F // [x^{i-1}y^j] mint lhs = 0, rhs = 0; for (auto&& [a, b, c]: dat) { if (a < i && b <= j) lhs += dp[i - a][j - b] * mint(i - a); if (a <= i && b <= j) rhs += dp[i - a][j - b] * c * mint(a); } dp[i][j] = (n * rhs - lhs) * inv(i); } } return dp; } #line 2 "/home/maspy/compro/library/nt/primetable.hpp" template vc primetable(int LIM) { ++LIM; const int S = 32768; static int done = 2; static vc primes = {2}, sieve(S + 1); if (done < LIM) { done = LIM; primes = {2}, sieve.assign(S + 1, 0); const int R = LIM / 2; primes.reserve(int(LIM / log(LIM) * 1.1)); vc> cp; for (int i = 3; i <= S; i += 2) { if (!sieve[i]) { cp.eb(i, i * i / 2); for (int j = i * i; j <= S; j += 2 * i) sieve[j] = 1; } } for (int L = 1; L <= R; L += S) { array block{}; for (auto& [p, idx]: cp) for (int i = idx; i < S + L; idx = (i += p)) block[i - L] = 1; FOR(i, min(S, R - L)) if (!block[i]) primes.eb((L + i) * 2 + 1); } } int k = LB(primes, LIM + 1); return {primes.begin(), primes.begin() + k}; } #line 3 "/home/maspy/compro/library/mod/powertable.hpp" // a^0, ..., a^N template vc powertable_1(mint a, ll N) { // table of a^i vc f(N + 1, 1); FOR(i, N) f[i + 1] = a * f[i]; return f; } // 0^e, ..., N^e template vc powertable_2(ll e, ll N) { auto primes = primetable(N); vc f(N + 1, 1); f[0] = mint(0).pow(e); for (auto&& p: primes) { if (p > N) break; mint xp = mint(p).pow(e); ll pp = p; while (pp <= N) { ll i = pp; while (i <= N) { f[i] *= xp; i += pp; } pp *= p; } } return f; } #line 7 "main.cpp" using mint = modint998; void solve() { LL(N, K); mint ANS = 0; auto POW = powertable_2(N + K, N); FOR(i, N + 1) { mint x = C(N, i); if ((N - i) & 1) x = -x; ANS += x * POW[i]; } ANS /= mint(N).pow(K); mint d = 1; FOR(i, 1, N + 1) d *= K + i; ANS /= d; print(ANS); } signed main() { int T = 1; // INT(T); FOR(T) solve(); return 0; }