This documentation is automatically generated by online-judge-tools/verification-helper
View the Project on GitHub maspypy/library
#define PROBLEM "https://yukicoder.me/problems/no/2166" #include "my_template.hpp" #include "other/io.hpp" #include "mod/modint.hpp" #include "poly/from_log_differentiation.hpp" #include "poly/multipoint.hpp" const int mod = 998244353; using mint = modint998; using poly = vc<mint>; using MAT = array<array<poly, 2>, 2>; struct Mono { using value_type = MAT; using X = value_type; static X op(X x, X y) { // これは Nlog^2N なので、雑で大丈夫 → そうでもない説 int nx = 0, ny = 0; FOR(i, 2) FOR(j, 2) chmax(nx, len(x[i][j])); FOR(i, 2) FOR(j, 2) chmax(ny, len(y[i][j])); int n = nx + ny - 1; int fft_len = 1; while (fft_len < n) fft_len *= 2; FOR(i, 2) FOR(j, 2) { x[i][j].resize(fft_len); ntt(x[i][j], false); } FOR(i, 2) FOR(j, 2) { y[i][j].resize(fft_len); ntt(y[i][j], false); } X z; FOR(i, 2) FOR(j, 2) z[i][j].resize(fft_len); FOR(i, 2) FOR(j, 2) FOR(k, 2) { FOR(p, fft_len) z[i][k][p] += x[i][j][p] * y[j][k][p]; } FOR(i, 2) FOR(j, 2) { ntt(z[i][j], true); z[i][j].resize(n); } return z; } static X unit() { MAT x; x[0][0] = x[1][1] = {mint(1)}; x[0][1] = x[1][0] = {}; return x; } static constexpr bool commute = 0; }; void solve_1(int Q) { VEC(pi, query, Q); auto make_mat = [&](ll K) -> MAT { MAT x; x[0][0] = {mint(-K - K), mint(2)}; // 2N-2K x[0][1] = {mint(-K * (K - 1) / 2), mint(K)}; // KN - K(K-1)/2 x[1][0] = {mint(1)}; x[1][1] = {}; return x; }; int MAX = 100'000; const int b_sz = 5010; const int b_num = ceil(MAX, b_sz) + 1; vvc<int> QID(b_num); FOR(q, Q) { auto [n, k] = query[q]; QID[k / b_sz].eb(q); } auto prod_range = [&](int L, int R) -> MAT { assert(L < R); vc<MAT> dat(R - L); FOR(i, R - L) dat[i] = make_mat(L + i); reverse(all(dat)); while (len(dat) > 1) { int n = len(dat); FOR(i, n) if (i % 2 == 1) { dat[i - 1] = Mono::op(dat[i - 1], dat[i]); } FOR(i, n) if (i % 2 == 0) dat[i / 2] = dat[i]; dat.resize(ceil(n, 2)); } return dat[0]; }; vc<mint> ANS(Q); MAT suffix_prod = Mono::unit(); FOR(b, b_num) { // suffix_prod に必要なものたちを ME する vc<mint> X; for (auto&& q: QID[b]) { X.eb(query[q].fi); } if (len(X)) { SubproductTree<mint> ST(X); auto Y0 = ST.evaluation(suffix_prod[0][0]); auto Y1 = ST.evaluation(suffix_prod[1][0]); FOR(t, len(X)) { int qid = QID[b][t]; auto [N, K] = query[qid]; N %= mod; pi p = {Y0[t].val, Y1[t].val}; FOR(k, b * b_sz, K) { ll c = k * (N + N - k + 1) / 2 % mod; p = {(2 * (N - k) * p.fi + c * p.se) % mod, p.fi}; } ANS[qid] = p.fi; } } suffix_prod = Mono::op(prod_range(b * b_sz, b * b_sz + b_sz), suffix_prod); } FOR(q, Q) print(ANS[q]); } mint solve_2(ll N, ll K) { if (K >= mod) return 0; assert(K <= mod); poly f = {mint(2 * N), mint(N)}; poly g = {mint(1), mint(2), inv<mint>(2)}; mint fa = [&]() -> mint { vc<mint> f = {1, 1}; return prefix_product_of_poly(f, K).val; }(); mint ANS = fa * from_log_differentiation_kth(K, f, g); return ANS; } void solve() { INT(T); if (T <= 10) { FOR(T) { LL(N, K); print(solve_2(N, K)); } return; } return solve_1(T); } signed main() { solve(); return 0; }
#line 1 "test/3_yukicoder/2166.test.cpp" #define PROBLEM "https://yukicoder.me/problems/no/2166" #line 1 "my_template.hpp" #if defined(LOCAL) #include <my_template_compiled.hpp> #else // https://codeforces.com/blog/entry/96344 #pragma GCC optimize("Ofast,unroll-loops") // いまの CF だとこれ入れると動かない? // #pragma GCC target("avx2,popcnt") #include <bits/stdc++.h> using namespace std; using ll = long long; using u8 = uint8_t; using u16 = uint16_t; using u32 = uint32_t; using u64 = uint64_t; using i128 = __int128; using u128 = unsigned __int128; using f128 = __float128; template <class T> constexpr T infty = 0; template <> constexpr int infty<int> = 1'010'000'000; template <> constexpr ll infty<ll> = 2'020'000'000'000'000'000; template <> constexpr u32 infty<u32> = infty<int>; template <> constexpr u64 infty<u64> = infty<ll>; template <> constexpr i128 infty<i128> = i128(infty<ll>) * 2'000'000'000'000'000'000; template <> constexpr double infty<double> = infty<ll>; template <> constexpr long double infty<long double> = infty<ll>; using pi = pair<ll, ll>; using vi = vector<ll>; template <class T> using vc = vector<T>; template <class T> using vvc = vector<vc<T>>; template <class T> using vvvc = vector<vvc<T>>; template <class T> using vvvvc = vector<vvvc<T>>; template <class T> using vvvvvc = vector<vvvvc<T>>; template <class T> using pq = priority_queue<T>; template <class T> using pqg = priority_queue<T, vector<T>, greater<T>>; #define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__)) #define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__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_sgn(int x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(u32 x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(ll x) { return (__builtin_parity(x) & 1 ? -1 : 1); } int popcnt_sgn(u64 x) { return (__builtin_parity(x) & 1 ? -1 : 1); } // (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 <typename T> T floor(T a, T b) { return a / b - (a % b && (a ^ b) < 0); } template <typename T> T ceil(T x, T y) { return floor(x + y - 1, y); } template <typename T> T bmod(T x, T y) { return x - y * floor(x, y); } template <typename T> pair<T, T> divmod(T x, T y) { T q = floor(x, y); return {q, x - q * y}; } template <typename T, typename U> T SUM(const vector<U> &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 <typename T> T POP(deque<T> &que) { T a = que.front(); que.pop_front(); return a; } template <typename T> T POP(pq<T> &que) { T a = que.top(); que.pop(); return a; } template <typename T> T POP(pqg<T> &que) { T a = que.top(); que.pop(); return a; } template <typename T> T POP(vc<T> &que) { T a = que.back(); que.pop_back(); return a; } template <typename F> 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 <typename F> 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 <class T, class S> inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template <class T, class S> inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } // ? は -1 vc<int> s_to_vi(const string &S, char first_char) { vc<int> A(S.size()); FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); } return A; } template <typename T, typename U> vector<T> cumsum(vector<U> &A, int off = 1) { int N = A.size(); vector<T> 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 <typename T> vector<int> argsort(const vector<T> &A) { vector<int> 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 <typename T> vc<T> rearrange(const vc<T> &A, const vc<int> &I) { vc<T> B(len(I)); FOR(i, len(I)) B[i] = A[I[i]]; return B; } template <typename T, typename... Vectors> void concat(vc<T> &first, const Vectors &... others) { vc<T> &res = first; (res.insert(res.end(), others.begin(), others.end()), ...); } #endif #line 1 "other/io.hpp" #define FASTIO #include <unistd.h> // 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 <typename T> void rd_real(T &x) { string s; rd(s); x = stod(s); } template <typename T> 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<T>::value || is_same_v<T, i128>) { if (c == '-') { minus = 1, c = ibuf[pil++]; } } x = 0; while ('0' <= c) { x = x * 10 + (c & 15), c = ibuf[pil++]; } if constexpr (is_signed<T>::value || is_same_v<T, i128>) { 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 <class T, class U> void rd(pair<T, U> &p) { return rd(p.first), rd(p.second); } template <size_t N = 0, typename T> void rd_tuple(T &t) { if constexpr (N < std::tuple_size<T>::value) { auto &x = std::get<N>(t); rd(x); rd_tuple<N + 1>(t); } } template <class... T> void rd(tuple<T...> &tpl) { rd_tuple(tpl); } template <size_t N = 0, typename T> void rd(array<T, N> &x) { for (auto &d: x) rd(d); } template <class T> void rd(vc<T> &x) { for (auto &d: x) rd(d); } void read() {} template <class H, class... T> 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 <typename T> 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 <typename T> 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 <class T, class U> void wt(const pair<T, U> val) { wt(val.first); wt(' '); wt(val.second); } template <size_t N = 0, typename T> void wt_tuple(const T t) { if constexpr (N < std::tuple_size<T>::value) { if constexpr (N > 0) { wt(' '); } const auto x = std::get<N>(t); wt(x); wt_tuple<N + 1>(t); } } template <class... T> void wt(tuple<T...> tpl) { wt_tuple(tpl); } template <class T, size_t S> void wt(const array<T, S> val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } template <class T> void wt(const vector<T> val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) wt(' '); wt(val[i]); } } void print() { wt('\n'); } template <class Head, class... Tail> void print(Head &&head, Tail &&... tail) { wt(head); if (sizeof...(Tail)) wt(' '); print(forward<Tail>(tail)...); } // gcc expansion. called automaticall after main. void __attribute__((destructor)) _d() { flush(); } } // namespace fastio using fastio::read; using fastio::print; using fastio::flush; #if defined(LOCAL) #define SHOW(...) SHOW_IMPL(__VA_ARGS__, SHOW6, SHOW5, SHOW4, SHOW3, SHOW2, SHOW1)(__VA_ARGS__) #define SHOW_IMPL(_1, _2, _3, _4, _5, _6, NAME, ...) NAME #define SHOW1(x) print(#x, "=", (x)), flush() #define SHOW2(x, y) print(#x, "=", (x), #y, "=", (y)), flush() #define SHOW3(x, y, z) print(#x, "=", (x), #y, "=", (y), #z, "=", (z)), flush() #define SHOW4(x, y, z, w) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w)), flush() #define SHOW5(x, y, z, w, v) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v)), flush() #define SHOW6(x, y, z, w, v, u) print(#x, "=", (x), #y, "=", (y), #z, "=", (z), #w, "=", (w), #v, "=", (v), #u, "=", (u)), flush() #else #define SHOW(...) #endif #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<type> name(size); \ read(name) #define VV(type, name, h, w) \ vector<vector<type>> name(h, vector<type>(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 2 "mod/modint_common.hpp" struct has_mod_impl { template <class T> static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{}); template <class T> static auto check(...) -> std::false_type; }; template <class T> class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {}; template <typename mint> mint inv(int n) { static const int mod = mint::get_mod(); static vector<mint> 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 <typename mint> mint fact(int n) { static const int mod = mint::get_mod(); assert(0 <= n && n < mod); static vector<mint> dat = {1, 1}; while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat))); return dat[n]; } template <typename mint> mint fact_inv(int n) { static vector<mint> dat = {1, 1}; if (n < 0) return mint(0); while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat))); return dat[n]; } template <class mint, class... Ts> mint fact_invs(Ts... xs) { return (mint(1) * ... * fact_inv<mint>(xs)); } template <typename mint, class Head, class... Tail> mint multinomial(Head &&head, Tail &&... tail) { return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...); } template <typename mint> mint C_dense(int n, int k) { assert(n >= 0); if (k < 0 || n < k) return 0; static vvc<mint> 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 <typename mint, bool large = false, bool dense = false> mint C(ll n, ll k) { assert(n >= 0); if (k < 0 || n < k) return 0; if constexpr (dense) return C_dense<mint>(n, k); if constexpr (!large) return multinomial<mint>(n, k, n - k); k = min(k, n - k); mint x(1); FOR(i, k) x *= mint(n - i); return x * fact_inv<mint>(k); } template <typename mint, bool large = false> mint C_inv(ll n, ll k) { assert(n >= 0); assert(0 <= k && k <= n); if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k); return mint(1) / C<mint, 1>(n, k); } // [x^d](1-x)^{-n} template <typename mint, bool large = false, bool dense = false> 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<mint, large, dense>(n + d - 1, d); } #line 3 "mod/modint.hpp" template <int mod> 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<int, int> 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 == 1004535809) return {21, 582313106}; if (mod == 1012924417) return {21, 368093570}; return {-1, -1}; } static constexpr bool can_ntt() { return ntt_info().fi != -1; } }; #ifdef FASTIO template <int mod> void rd(modint<mod> &x) { fastio::rd(x.val); x.val %= mod; // assert(0 <= x.val && x.val < mod); } template <int mod> void wt(modint<mod> x) { fastio::wt(x.val); } #endif using modint107 = modint<1000000007>; using modint998 = modint<998244353>; #line 2 "poly/from_log_differentiation.hpp" #line 3 "linalg/matrix_mul.hpp" template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr> vc<vc<T>> matrix_mul(const vc<vc<T>>& A, const vc<vc<T>>& B, int N1 = -1, int N2 = -1, int N3 = -1) { if (N1 == -1) { N1 = len(A), N2 = len(B), N3 = len(B[0]); } vv(u32, b, N3, N2); FOR(i, N2) FOR(j, N3) b[j][i] = B[i][j].val; vv(T, C, N1, N3); if ((T::get_mod() < (1 << 30)) && N2 <= 16) { FOR(i, N1) FOR(j, N3) { u64 sm = 0; FOR(m, N2) sm += u64(A[i][m].val) * b[j][m]; C[i][j] = sm; } } else { FOR(i, N1) FOR(j, N3) { u128 sm = 0; FOR(m, N2) sm += u64(A[i][m].val) * b[j][m]; C[i][j] = T::raw(sm % (T::get_mod())); } } return C; } template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr> vc<vc<T>> matrix_mul(const vc<vc<T>>& A, const vc<vc<T>>& B, int N1 = -1, int N2 = -1, int N3 = -1) { if (N1 == -1) { N1 = len(A), N2 = len(B), N3 = len(B[0]); } vv(T, b, N2, N3); FOR(i, N2) FOR(j, N3) b[j][i] = B[i][j]; vv(T, C, N1, N3); FOR(n, N1) FOR(m, N2) FOR(k, N3) C[n][k] += A[n][m] * b[k][m]; return C; } // square-matrix defined as array template <class T, int N, typename enable_if<has_mod<T>::value>::type* = nullptr> array<array<T, N>, N> matrix_mul(const array<array<T, N>, N>& A, const array<array<T, N>, N>& B) { array<array<T, N>, N> C{}; if ((T::get_mod() < (1 << 30)) && N <= 16) { FOR(i, N) FOR(k, N) { u64 sm = 0; FOR(j, N) sm += u64(A[i][j].val) * (B[j][k].val); C[i][k] = sm; } } else { FOR(i, N) FOR(k, N) { u128 sm = 0; FOR(j, N) sm += u64(A[i][j].val) * (B[j][k].val); C[i][k] = sm; } } return C; } // square-matrix defined as array template <class T, int N, typename enable_if<!has_mod<T>::value>::type* = nullptr> array<array<T, N>, N> matrix_mul(const array<array<T, N>, N>& A, const array<array<T, N>, N>& B) { array<array<T, N>, N> C{}; FOR(i, N) FOR(j, N) FOR(k, N) C[i][k] += A[i][j] * B[j][k]; return C; } #line 2 "alg/monoid/mul.hpp" template <class T> struct Monoid_Mul { using value_type = T; using X = T; static constexpr X op(const X &x, const X &y) noexcept { return x * y; } static constexpr X inverse(const X &x) noexcept { return X(1) / x; } static constexpr X unit() { return X(1); } static constexpr bool commute = true; }; #line 1 "ds/sliding_window_aggregation.hpp" template <class Monoid> struct Sliding_Window_Aggregation { using X = typename Monoid::value_type; using value_type = X; int sz = 0; vc<X> dat; vc<X> cum_l; X cum_r; Sliding_Window_Aggregation() : cum_l({Monoid::unit()}), cum_r(Monoid::unit()) {} int size() { return sz; } void push(X x) { ++sz; cum_r = Monoid::op(cum_r, x); dat.eb(x); } void pop() { --sz; cum_l.pop_back(); if (len(cum_l) == 0) { cum_l = {Monoid::unit()}; cum_r = Monoid::unit(); while (len(dat) > 1) { cum_l.eb(Monoid::op(dat.back(), cum_l.back())); dat.pop_back(); } dat.pop_back(); } } X lprod() { return cum_l.back(); } X rprod() { return cum_r; } X prod() { return Monoid::op(cum_l.back(), cum_r); } }; // 定数倍は目に見えて遅くなるので、queue でよいときは使わない template <class Monoid> struct SWAG_deque { using X = typename Monoid::value_type; using value_type = X; int sz; vc<X> dat_l, dat_r; vc<X> cum_l, cum_r; SWAG_deque() : sz(0), cum_l({Monoid::unit()}), cum_r({Monoid::unit()}) {} int size() { return sz; } void push_back(X x) { ++sz; dat_r.eb(x); cum_r.eb(Monoid::op(cum_r.back(), x)); } void push_front(X x) { ++sz; dat_l.eb(x); cum_l.eb(Monoid::op(x, cum_l.back())); } void push(X x) { push_back(x); } void clear() { sz = 0; dat_l.clear(), dat_r.clear(); cum_l = {Monoid::unit()}, cum_r = {Monoid::unit()}; } void pop_front() { if (sz == 1) return clear(); if (dat_l.empty()) rebuild(); --sz; dat_l.pop_back(); cum_l.pop_back(); } void pop_back() { if (sz == 1) return clear(); if (dat_r.empty()) rebuild(); --sz; dat_r.pop_back(); cum_r.pop_back(); } void pop() { pop_front(); } X front() { if (len(dat_l)) return dat_l.back(); return dat_r[0]; } X lprod() { return cum_l.back(); } X rprod() { return cum_r.back(); } X prod() { return Monoid::op(cum_l.back(), cum_r.back()); } X prod_all() { return prod(); } private: void rebuild() { vc<X> X; reverse(all(dat_l)); concat(X, dat_l, dat_r); clear(); int m = len(X) / 2; FOR_R(i, m) push_front(X[i]); FOR(i, m, len(X)) push_back(X[i]); assert(sz == len(X)); } }; #line 2 "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 2 "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 <typename T, u32 p0, u32 p1> T CRT2(u64 a0, u64 a1) { static_assert(p0 < p1); static constexpr u64 x0_1 = mod_pow_constexpr(p0, p1 - 2, p1); u64 c = (a1 - a0 + p1) * x0_1 % p1; return a0 + c * p0; } template <typename T, u32 p0, u32 p1, u32 p2> T CRT3(u64 a0, u64 a1, u64 a2) { static_assert(p0 < p1 && p1 < p2); static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1); static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2); static constexpr u64 p01 = u64(p0) * p1; u64 c = (a1 - a0 + p1) * x1 % p1; u64 ans_1 = a0 + c * p0; c = (a2 - ans_1 % p2 + p2) * x2 % p2; return T(ans_1) + T(c) * T(p01); } template <typename T, u32 p0, u32 p1, u32 p2, u32 p3> T CRT4(u64 a0, u64 a1, u64 a2, u64 a3) { static_assert(p0 < p1 && p1 < p2 && p2 < p3); static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1); static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2); static constexpr u64 x3 = mod_pow_constexpr(u64(p0) * p1 % p3 * p2 % p3, p3 - 2, p3); static constexpr u64 p01 = u64(p0) * p1; u64 c = (a1 - a0 + p1) * x1 % p1; u64 ans_1 = a0 + c * p0; c = (a2 - ans_1 % p2 + p2) * x2 % p2; u128 ans_2 = ans_1 + c * static_cast<u128>(p01); c = (a3 - ans_2 % p3 + p3) * x3 % p3; return T(ans_2) + T(c) * T(p01) * T(p2); } template <typename T, u32 p0, u32 p1, u32 p2, u32 p3, u32 p4> T CRT5(u64 a0, u64 a1, u64 a2, u64 a3, u64 a4) { static_assert(p0 < p1 && p1 < p2 && p2 < p3 && p3 < p4); static constexpr u64 x1 = mod_pow_constexpr(p0, p1 - 2, p1); static constexpr u64 x2 = mod_pow_constexpr(u64(p0) * p1 % p2, p2 - 2, p2); static constexpr u64 x3 = mod_pow_constexpr(u64(p0) * p1 % p3 * p2 % p3, p3 - 2, p3); static constexpr u64 x4 = mod_pow_constexpr(u64(p0) * p1 % p4 * p2 % p4 * p3 % p4, p4 - 2, p4); static constexpr u64 p01 = u64(p0) * p1; static constexpr u64 p23 = u64(p2) * p3; u64 c = (a1 - a0 + p1) * x1 % p1; u64 ans_1 = a0 + c * p0; c = (a2 - ans_1 % p2 + p2) * x2 % p2; u128 ans_2 = ans_1 + c * static_cast<u128>(p01); c = static_cast<u64>(a3 - ans_2 % p3 + p3) * x3 % p3; u128 ans_3 = ans_2 + static_cast<u128>(c * p2) * p01; c = static_cast<u64>(a4 - ans_3 % p4 + p4) * x4 % p4; return T(ans_3) + T(c) * T(p01) * T(p23); } #line 2 "poly/convolution_naive.hpp" template <class T, typename enable_if<!has_mod<T>::value>::type* = nullptr> vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) { int n = int(a.size()), m = int(b.size()); if (n > m) return convolution_naive<T>(b, a); if (n == 0) return {}; vector<T> ans(n + m - 1); FOR(i, n) FOR(j, m) ans[i + j] += a[i] * b[j]; return ans; } template <class T, typename enable_if<has_mod<T>::value>::type* = nullptr> vc<T> convolution_naive(const vc<T>& a, const vc<T>& b) { int n = int(a.size()), m = int(b.size()); if (n > m) return convolution_naive<T>(b, a); if (n == 0) return {}; vc<T> 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 "poly/convolution_karatsuba.hpp" // 任意の環でできる template <typename T> vc<T> convolution_karatsuba(const vc<T>& f, const vc<T>& 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<T> 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<T> a = convolution_karatsuba(f1, g1); vc<T> b = convolution_karatsuba(f2, g2); FOR(i, len(f2)) f1[i] += f2[i]; FOR(i, len(g2)) g1[i] += g2[i]; vc<T> c = convolution_karatsuba(f1, g1); vc<T> 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 "poly/ntt.hpp" template <class mint> void ntt(vector<mint>& a, bool inverse) { assert(mint::can_ntt()); const int rank2 = mint::ntt_info().fi; const int mod = mint::get_mod(); static array<mint, 30> root, iroot; static array<mint, 30> rate2, irate2; static array<mint, 30> rate3, irate3; assert(rank2 != -1 && len(a) <= (1 << max(0, rank2))); 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 8 "poly/convolution.hpp" template <class mint> vector<mint> convolution_ntt(vector<mint> a, vector<mint> 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 <typename mint> vector<mint> convolution_garner(const vector<mint>& a, const vector<mint>& 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<p0>; using mint1 = modint<p1>; using mint2 = modint<p2>; vc<mint0> a0(n), b0(m); vc<mint1> a1(n), b1(m); vc<mint2> 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<mint0>(a0, b0); auto c1 = convolution_ntt<mint1>(a1, b1); auto c2 = convolution_ntt<mint2>(a2, b2); vc<mint> c(len(c0)); FOR(i, n + m - 1) { c[i] = CRT3<mint, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val); } return c; } vector<ll> convolution(vector<ll> a, vector<ll> b) { int n = len(a), m = len(b); if (!n || !m) return {}; if (min(n, m) <= 2500) return convolution_naive(a, b); ll mi_a = MIN(a), mi_b = MIN(b); for (auto& x: a) x -= mi_a; for (auto& x: b) x -= mi_b; assert(MAX(a) * MAX(b) <= 1e18); auto Ac = cumsum<ll>(a), Bc = cumsum<ll>(b); vi res(n + m - 1); for (int k = 0; k < n + m - 1; ++k) { int s = max(0, k - m + 1); int t = min(n, k + 1); res[k] += (t - s) * mi_a * mi_b; res[k] += mi_a * (Bc[k - s + 1] - Bc[k - t + 1]); res[k] += mi_b * (Ac[t] - Ac[s]); } static constexpr u32 MOD1 = 1004535809; static constexpr u32 MOD2 = 1012924417; using mint1 = modint<MOD1>; using mint2 = modint<MOD2>; vc<mint1> a1(n), b1(m); vc<mint2> a2(n), b2(m); FOR(i, n) a1[i] = a[i], a2[i] = a[i]; FOR(i, m) b1[i] = b[i], b2[i] = b[i]; auto c1 = convolution_ntt<mint1>(a1, b1); auto c2 = convolution_ntt<mint2>(a2, b2); FOR(i, n + m - 1) { res[i] += CRT2<u64, MOD1, MOD2>(c1[i].val, c2[i].val); } return res; } template <typename mint> vc<mint> convolution(const vc<mint>& a, const vc<mint>& b) { int n = len(a), m = len(b); if (!n || !m) return {}; if (mint::can_ntt()) { if (min(n, m) <= 50) return convolution_karatsuba<mint>(a, b); return convolution_ntt(a, b); } if (min(n, m) <= 200) return convolution_karatsuba<mint>(a, b); return convolution_garner(a, b); } #line 5 "poly/lagrange_interpolate_iota.hpp" // Input: f(0), ..., f(n-1) and c. Return: f(c) template <typename T, typename enable_if<has_mod<T>::value>::type * = nullptr> T lagrange_interpolate_iota(vc<T> &f, T c) { int n = len(f); if (int(c.val) < n) return f[c.val]; auto a = f; FOR(i, n) { a[i] = a[i] * fact_inv<T>(i) * fact_inv<T>(n - 1 - i); if ((n - 1 - i) & 1) a[i] = -a[i]; } vc<T> lp(n + 1), rp(n + 1); lp[0] = rp[n] = 1; FOR(i, n) lp[i + 1] = lp[i] * (c - i); FOR_R(i, n) rp[i] = rp[i + 1] * (c - i); T ANS = 0; FOR(i, n) ANS += a[i] * lp[i] * rp[i + 1]; return ANS; } // mod じゃない場合。かなり低次の多項式を想定している。O(n^2) // Input: f(0), ..., f(n-1) and c. Return: f(c) template <typename T, typename enable_if<!has_mod<T>::value>::type * = nullptr> T lagrange_interpolate_iota(vc<T> &f, T c) { const int LIM = 10; int n = len(f); assert(n < LIM); // (-1)^{i-j} binom(i,j) static vvc<int> C; if (C.empty()) { C.assign(LIM, vc<int>(LIM)); C[0][0] = 1; FOR(n, 1, LIM) FOR(k, n + 1) { C[n][k] += C[n - 1][k]; if (k) C[n][k] += C[n - 1][k - 1]; } FOR(n, LIM) FOR(k, n + 1) if ((n + k) % 2) C[n][k] = -C[n][k]; } // f(x) = sum a_i binom(x,i) vc<T> a(n); FOR(i, n) FOR(j, i + 1) { a[i] += f[j] * C[i][j]; } T res = 0; T b = 1; FOR(i, n) { res += a[i] * b; b = b * (c - i) / (1 + i); } return res; } // Input: f(0), ..., f(n-1) and c, m // Return: f(c), f(c+1), ..., f(c+m-1) // Complexity: M(n, n + m) template <typename mint> vc<mint> lagrange_interpolate_iota(vc<mint> &f, mint c, int m) { if (m <= 60) { vc<mint> ANS(m); FOR(i, m) ANS[i] = lagrange_interpolate_iota(f, c + mint(i)); return ANS; } ll n = len(f); auto a = f; FOR(i, n) { a[i] = a[i] * fact_inv<mint>(i) * fact_inv<mint>(n - 1 - i); if ((n - 1 - i) & 1) a[i] = -a[i]; } // x = c, c+1, ... に対して a0/x + a1/(x-1) + ... を求めておく vc<mint> b(n + m - 1); FOR(i, n + m - 1) b[i] = mint(1) / (c + mint(i - n + 1)); a = convolution(a, b); Sliding_Window_Aggregation<Monoid_Mul<mint>> swag; vc<mint> ANS(m); ll L = 0, R = 0; FOR(i, m) { while (L < i) { swag.pop(), ++L; } while (R - L < n) { swag.push(c + mint((R++) - n + 1)); } auto coef = swag.prod(); if (coef == 0) { ANS[i] = f[(c + i).val]; } else { ANS[i] = a[i + n - 1] * coef; } } return ANS; } #line 4 "poly/prefix_product_of_poly.hpp" // A[k-1]...A[0] を計算する // アルゴリズム参考:https://github.com/noshi91/n91lib_rs/blob/master/src/algorithm/polynomial_matrix_prod.rs // 実装参考:https://nyaannyaan.github.io/library/matrix/polynomial-matrix-prefix-prod.hpp template <typename T> vc<vc<T>> prefix_product_of_poly_matrix(vc<vc<vc<T>>>& A, ll k) { int n = len(A); using MAT = vc<vc<T>>; auto shift = [&](vc<MAT>& G, T x) -> vc<MAT> { int d = len(G); vvv(T, H, d, n, n); FOR(i, n) FOR(j, n) { vc<T> g(d); FOR(l, d) g[l] = G[l][i][j]; auto h = lagrange_interpolate_iota(g, x, d); FOR(l, d) H[l][i][j] = h[l]; } return H; }; auto evaluate = [&](vc<T>& f, T x) -> T { T res = 0; T p = 1; FOR(i, len(f)) { res += f[i] * p; p *= x; } return res; }; ll deg = 1; FOR(i, n) FOR(j, n) chmax(deg, len(A[i][j]) - 1); vc<MAT> G(deg + 1); ll v = 1; while (deg * v * v < k) v *= 2; T iv = T(1) / T(v); FOR(i, len(G)) { T x = T(v) * T(i); vv(T, mat, n, n); FOR(j, n) FOR(k, n) mat[j][k] = evaluate(A[j][k], x); G[i] = mat; } for (ll w = 1; w != v; w *= 2) { T W = w; auto G1 = shift(G, W * iv); auto G2 = shift(G, (W * T(deg) * T(v) + T(v)) * iv); auto G3 = shift(G, (W * T(deg) * T(v) + T(v) + W) * iv); FOR(i, w * deg + 1) { G[i] = matrix_mul(G1[i], G[i]); G2[i] = matrix_mul(G3[i], G2[i]); } copy(G2.begin(), G2.end() - 1, back_inserter(G)); } vv(T, res, n, n); FOR(i, n) res[i][i] = 1; ll i = 0; while (i + v <= k) res = matrix_mul(G[i / v], res), i += v; while (i < k) { vv(T, mat, n, n); FOR(j, n) FOR(k, n) mat[j][k] = evaluate(A[j][k], i); res = matrix_mul(mat, res); ++i; } return res; } // f[k-1]...f[0] を計算する template <typename T> T prefix_product_of_poly(vc<T>& f, ll k) { vc<vc<vc<T>>> A(1); A[0].resize(1); A[0][0] = f; auto res = prefix_product_of_poly_matrix(A, k); return res[0][0]; } #line 2 "seq/kth_term_of_p_recursive.hpp" // a0, ..., a_{r-1} および f_0, ..., f_r を与える // a_r f_0(0) + a_{r-1}f_1(0) + ... = 0 // a_{r+1} f_0(1) + a_{r}f_1(1) + ... = 0 template <typename T> T kth_term_of_p_recursive(vc<T> a, vc<vc<T>>& fs, ll k) { int r = len(a); assert(len(fs) == r + 1); if (k < r) return a[k]; vc<vc<vc<T>>> A; A.resize(r); FOR(i, r) A[i].resize(r); FOR(i, r) { // A[0][i] = -fs[i + 1]; for (auto&& x: fs[i + 1]) A[0][i].eb(-x); } FOR3(i, 1, r) A[i][i - 1] = fs[0]; vc<T> den = fs[0]; auto res = prefix_product_of_poly_matrix(A, k - r + 1); reverse(all(a)); T ANS = 0; FOR(j, r) ANS += res[0][j] * a[j]; ANS /= prefix_product_of_poly(den, k - r + 1); return ANS; } #line 4 "poly/from_log_differentiation.hpp" // 対数微分 F'/F = a(x)/b(x) から F を復元する。 // a, b が sparse であれば、O(N(K1+K2)) 時間でできる // [0, N] を計算 template <typename mint> vc<mint> from_log_differentiation(int N, const vc<mint>& a, const vc<mint>& b) { assert(b[0] == mint(1)); using P = pair<int, mint>; vc<P> dat_a, dat_b; FOR(i, len(a)) if (a[i] != mint(0)) dat_a.eb(i, a[i]); FOR(i, 1, len(b)) if (b[i] != mint(0)) dat_b.eb(i, b[i]); vc<mint> f(N + 1); vc<mint> df(N); f[0] = mint(1); FOR(n, N) { mint v = 0; for (auto&& [i, bi]: dat_b) { if (i > n) break; v -= bi * df[n - i]; } for (auto&& [i, ai]: dat_a) { if (i > n) break; v += ai * f[n - i]; } df[n] = v; f[n + 1] = df[n] * inv<mint>(n + 1); } return f; } // F'/F = a/b の解の、[x^K]F を求める。右辺は低次の有理式。 template <typename mint> mint from_log_differentiation_kth(int K, vc<mint>& a, vc<mint>& b) { assert(b[0] == mint(1)); int r = max(len(a), len(b) - 1); vvc<mint> c(r + 1); FOR(i, r + 1) { mint c0 = 0, c1 = 0; if (i < len(b)) c0 += mint(r - i) * b[i]; if (i < len(b)) c1 += b[i]; if (0 <= i - 1 && i - 1 < len(b)) c0 -= a[i - 1]; c[i] = {c0, c1}; } auto f = from_log_differentiation(r - 1, a, b); mint ANS = kth_term_of_p_recursive(f, c, K); return ANS; } #line 2 "poly/multipoint.hpp" #line 2 "poly/middle_product.hpp" #line 6 "poly/middle_product.hpp" // n, m 次多項式 (n>=m) a, b → n-m 次多項式 c // c[i] = sum_j b[j]a[i+j] template <typename mint> vc<mint> middle_product(vc<mint>& a, vc<mint>& b) { assert(len(a) >= len(b)); if (b.empty()) return vc<mint>(len(a) - len(b) + 1); if (min(len(b), len(a) - len(b) + 1) <= 60) { return middle_product_naive(a, b); } if (!(mint::can_ntt())) { return middle_product_garner(a, b); } else { int n = 1 << __lg(2 * len(a) - 1); vc<mint> fa(n), fb(n); copy(a.begin(), a.end(), fa.begin()); copy(b.rbegin(), b.rend(), fb.begin()); ntt(fa, 0), ntt(fb, 0); FOR(i, n) fa[i] *= fb[i]; ntt(fa, 1); fa.resize(len(a)); fa.erase(fa.begin(), fa.begin() + len(b) - 1); return fa; } } template <typename mint> vc<mint> middle_product_garner(vc<mint>& a, vc<mint> 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<p0>; using mint1 = modint<p1>; using mint2 = modint<p2>; vc<mint0> a0(n), b0(m); vc<mint1> a1(n), b1(m); vc<mint2> 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 = middle_product<mint0>(a0, b0); auto c1 = middle_product<mint1>(a1, b1); auto c2 = middle_product<mint2>(a2, b2); vc<mint> c(len(c0)); FOR(i, n - m + 1) { c[i] = CRT3<mint, p0, p1, p2>(c0[i].val, c1[i].val, c2[i].val); } return c; } template <typename mint> vc<mint> middle_product_naive(vc<mint>& a, vc<mint>& b) { vc<mint> res(len(a) - len(b) + 1); FOR(i, len(res)) FOR(j, len(b)) res[i] += b[j] * a[i + j]; return res; } #line 2 "mod/all_inverse.hpp" template <typename mint> vc<mint> all_inverse(vc<mint>& X) { for (auto&& x: X) assert(x != mint(0)); int N = len(X); vc<mint> res(N + 1); res[0] = mint(1); FOR(i, N) res[i + 1] = res[i] * X[i]; mint t = res.back().inverse(); res.pop_back(); FOR_R(i, N) { res[i] *= t; t *= X[i]; } return res; } #line 2 "poly/fps_div.hpp" #line 2 "poly/count_terms.hpp" template<typename mint> int count_terms(const vc<mint>& f){ int t = 0; FOR(i, len(f)) if(f[i] != mint(0)) ++t; return t; } #line 4 "poly/fps_inv.hpp" template <typename mint> vc<mint> fps_inv_sparse(const vc<mint>& f) { int N = len(f); vc<pair<int, mint>> dat; FOR(i, 1, N) if (f[i] != mint(0)) dat.eb(i, f[i]); vc<mint> 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 <typename mint> vc<mint> fps_inv_dense_ntt(const vc<mint>& F) { vc<mint> G = {mint(1) / F[0]}; ll N = len(F), n = 1; G.reserve(N); while (n < N) { vc<mint> 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 <typename mint> vc<mint> fps_inv_dense(const vc<mint>& F) { if (mint::can_ntt()) return fps_inv_dense_ntt(F); const int N = len(F); vc<mint> R = {mint(1) / F[0]}; vc<mint> p; int m = 1; while (m < N) { p = convolution(R, R); p.resize(m + m); vc<mint> 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 <typename mint> vc<mint> fps_inv(const vc<mint>& f) { assert(f[0] != mint(0)); int n = count_terms(f); int t = (mint::can_ntt() ? 160 : 820); return (n <= t ? fps_inv_sparse<mint>(f) : fps_inv_dense<mint>(f)); } #line 5 "poly/fps_div.hpp" // f/g. f の長さで出力される. template <typename mint, bool SPARSE = false> vc<mint> fps_div(vc<mint> f, vc<mint> g) { if (SPARSE || count_terms(g) < 200) return fps_div_sparse(f, g); int n = len(f); g.resize(n); g = fps_inv<mint>(g); f = convolution(f, g); f.resize(n); return f; } // f/g ただし g は sparse template <typename mint> vc<mint> fps_div_sparse(vc<mint> f, vc<mint>& g) { if (g[0] != mint(1)) { mint cf = g[0].inverse(); for (auto&& x: f) x *= cf; for (auto&& x: g) x *= cf; } vc<pair<int, mint>> dat; FOR(i, 1, len(g)) if (g[i] != mint(0)) dat.eb(i, -g[i]); FOR(i, len(f)) { for (auto&& [j, x]: dat) { if (i >= j) f[i] += x * f[i - j]; } } return f; } #line 2 "poly/ntt_doubling.hpp" #line 4 "poly/ntt_doubling.hpp" // 2^k 次多項式の長さ 2^k が与えられるので 2^k+1 にする template <typename mint, bool transposed = false> void ntt_doubling(vector<mint>& a) { static array<mint, 30> root; static bool prepared = 0; if (!prepared) { prepared = 1; const int rank2 = mint::ntt_info().fi; root[rank2] = mint::ntt_info().se; FOR_R(i, rank2) { root[i] = root[i + 1] * root[i + 1]; } } if constexpr (!transposed) { const int M = (int)a.size(); auto b = a; ntt(b, 1); mint r = 1, zeta = root[topbit(2 * M)]; FOR(i, M) b[i] *= r, r *= zeta; ntt(b, 0); copy(begin(b), end(b), back_inserter(a)); } else { const int M = len(a) / 2; vc<mint> tmp = {a.begin(), a.begin() + M}; a = {a.begin() + M, a.end()}; transposed_ntt(a, 0); mint r = 1, zeta = root[topbit(2 * M)]; FOR(i, M) a[i] *= r, r *= zeta; transposed_ntt(a, 1); FOR(i, M) a[i] += tmp[i]; } } #line 2 "poly/transposed_ntt.hpp" template <class mint> void transposed_ntt(vector<mint>& a, bool inverse) { assert(mint::can_ntt()); const int rank2 = mint::ntt_info().fi; const int mod = mint::get_mod(); static array<mint, 30> root, iroot; static array<mint, 30> rate2, irate2; static array<mint, 30> rate3, irate3; assert(rank2 != -1 && len(a) <= (1 << max(0, rank2))); 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 = h; while (len > 0) { if (len == 1) { int p = 1 << (h - len); mint rot = 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) * rot.val; } rot *= rate2[topbit(~s & -~s)]; } len--; } else { int p = 1 << (h - len); mint rot = 1, imag = root[2]; FOR(s, (1 << (len - 2))) { int offset = s << (h - len + 2); mint rot2 = rot * rot; mint rot3 = rot2 * rot; 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) * imag.val % mod; a[i + offset] = a0 + a1 + a2 + a3; a[i + offset + 1 * p] = (a0 + mod - a1 + x) * rot.val; a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * rot2.val; a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * rot3.val; } rot *= rate3[topbit(~s & -~s)]; } len -= 2; } } } else { mint coef = mint(1) / mint(len(a)); FOR(i, len(a)) a[i] *= coef; int len = 0; while (len < h) { if (len == h - 1) { int p = 1 << (h - len - 1); mint irot = 1; FOR(s, 1 << len) { int offset = s << (h - len); FOR(i, p) { auto l = a[i + offset]; auto r = a[i + offset + p] * irot; a[i + offset] = l + r; a[i + offset + p] = l - r; } irot *= irate2[topbit(~s & -~s)]; } len++; } else { int p = 1 << (h - len - 2); mint irot = 1, iimag = iroot[2]; for (int s = 0; s < (1 << len); s++) { mint irot2 = irot * irot; mint irot3 = irot2 * irot; 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) * irot.val; u64 a2 = u64(a[i + offset + 2 * p].val) * irot2.val; u64 a3 = u64(a[i + offset + 3 * p].val) * irot3.val; u64 a1na3imag = (a1 + mod2 - a3) % mod * iimag.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); } irot *= irate3[topbit(~s & -~s)]; } len += 2; } } } } #line 8 "poly/multipoint.hpp" template <typename mint> struct SubproductTree { int m; int sz; vc<vc<mint>> T; SubproductTree(const vc<mint>& x) { m = len(x); sz = 1; while (sz < m) sz *= 2; T.resize(2 * sz); FOR(i, sz) T[sz + i] = {1, (i < m ? -x[i] : 0)}; FOR3_R(i, 1, sz) T[i] = convolution(T[2 * i], T[2 * i + 1]); } vc<mint> evaluation(vc<mint> f) { int n = len(f); if (n == 0) return vc<mint>(m, mint(0)); f.resize(2 * n - 1); vc<vc<mint>> g(2 * sz); g[1] = T[1]; g[1].resize(n); g[1] = fps_inv(g[1]); g[1] = middle_product(f, g[1]); g[1].resize(sz); FOR3(i, 1, sz) { g[2 * i] = middle_product(g[i], T[2 * i + 1]); g[2 * i + 1] = middle_product(g[i], T[2 * i]); } vc<mint> vals(m); FOR(i, m) vals[i] = g[sz + i][0]; return vals; } vc<mint> interpolation(vc<mint>& y) { assert(len(y) == m); vc<mint> a(m); FOR(i, m) a[i] = T[1][m - i - 1] * (i + 1); a = evaluation(a); vc<vc<mint>> t(2 * sz); FOR(i, sz) t[sz + i] = {(i < m ? y[i] / a[i] : 0)}; FOR3_R(i, 1, sz) { t[i] = convolution(t[2 * i], T[2 * i + 1]); auto tt = convolution(t[2 * i + 1], T[2 * i]); FOR(k, len(t[i])) t[i][k] += tt[k]; } t[1].resize(m); reverse(all(t[1])); return t[1]; } }; template <typename mint> vc<mint> multipoint_evaluation_ntt(vc<mint> f, vc<mint> point) { using poly = vc<mint>; int n = 1, k = 0; while (n < len(point)) n *= 2, ++k; vv(mint, F, k + 1, 2 * n); FOR(i, len(point)) F[0][2 * i] = -point[i]; FOR(d, k) { int b = 1 << d; for (int L = 0; L < 2 * n; L += 4 * b) { poly f1 = {F[d].begin() + L, F[d].begin() + L + b}; poly f2 = {F[d].begin() + L + 2 * b, F[d].begin() + L + 3 * b}; ntt_doubling(f1), ntt_doubling(f2); FOR(i, b) f1[i] += 1, f2[i] += 1; FOR(i, b, 2 * b) f1[i] -= 1, f2[i] -= 1; copy(all(f1), F[d].begin() + L); copy(all(f2), F[d].begin() + L + 2 * b); FOR(i, 2 * b) { F[d + 1][L + i] = f1[i] * f2[i] - 1; } } } vc<mint> P = {F[k].begin(), F[k].begin() + n}; ntt(P, 1), P.eb(1), reverse(all(P)), P.resize(len(f)), P = fps_inv<mint>(P); f.resize(n + len(P) - 1), f = middle_product<mint>(f, P), reverse(all(f)); transposed_ntt(f, 1); vc<mint>& G = f; FOR_R(d, k) { vc<mint> nxt_G(n); int b = 1 << d; for (int L = 0; L < n; L += 2 * b) { vc<mint> g1(2 * b), g2(2 * b); FOR(i, 2 * b) { g1[i] = G[L + i] * F[d][2 * L + 2 * b + i]; } FOR(i, 2 * b) { g2[i] = G[L + i] * F[d][2 * L + i]; } ntt_doubling<mint, true>(g1), ntt_doubling<mint, true>(g2); FOR(i, b) { nxt_G[L + i] = g1[i], nxt_G[L + b + i] = g2[i]; } } swap(G, nxt_G); } G.resize(len(point)); return G; } template <typename mint> vc<mint> multipoint_eval(vc<mint>& f, vc<mint>& x) { if (x.empty()) return {}; if (mint::can_ntt()) return multipoint_evaluation_ntt(f, x); SubproductTree<mint> F(x); return F.evaluation(f); } template <typename mint> vc<mint> multipoint_interpolate(vc<mint>& x, vc<mint>& y) { if (x.empty()) return {}; SubproductTree<mint> F(x); return F.interpolation(y); } // calculate f(ar^k) for 0 <= k < m template <typename mint> vc<mint> multipoint_eval_on_geom_seq(vc<mint> f, mint a, mint r, int m) { const int n = len(f); if (m == 0) return {}; auto eval = [&](mint x) -> mint { mint fx = 0; mint pow = 1; FOR(i, n) fx += f[i] * pow, pow *= x; return fx; }; if (r == mint(0)) { vc<mint> res(m); FOR(i, 1, m) res[i] = f[0]; res[0] = eval(a); return res; } if (n < 60 || m < 60) { vc<mint> res(m); FOR(i, m) res[i] = eval(a), a *= r; return res; } assert(r != mint(0)); // a == 1 に帰着 mint pow_a = 1; FOR(i, n) f[i] *= pow_a, pow_a *= a; auto calc = [&](mint r, int m) -> vc<mint> { // r^{t_i} の計算 vc<mint> res(m); mint pow = 1; res[0] = 1; FOR(i, m - 1) { res[i + 1] = res[i] * pow; pow *= r; } return res; }; vc<mint> A = calc(r, n + m - 1), B = calc(r.inverse(), max(n, m)); FOR(i, n) f[i] *= B[i]; f = middle_product(A, f); FOR(i, m) f[i] *= B[i]; return f; } // Y[i] = f(ar^i) template <typename mint> vc<mint> multipoint_interpolate_on_geom_seq(vc<mint> Y, mint a, mint r) { const int n = len(Y); if (n == 0) return {}; if (n == 1) return {Y[0]}; assert(r != mint(0)); mint ir = r.inverse(); vc<mint> POW(n + n - 1), tPOW(n + n - 1); POW[0] = tPOW[0] = mint(1); FOR(i, n + n - 2) POW[i + 1] = POW[i] * r, tPOW[i + 1] = tPOW[i] * POW[i]; vc<mint> iPOW(n + n - 1), itPOW(n + n - 1); iPOW[0] = itPOW[0] = mint(1); FOR(i, n) iPOW[i + 1] = iPOW[i] * ir, itPOW[i + 1] = itPOW[i] * iPOW[i]; // prod_[1,i] 1-r^k vc<mint> S(n); S[0] = mint(1); FOR(i, 1, n) S[i] = S[i - 1] * (mint(1) - POW[i]); vc<mint> iS = all_inverse<mint>(S); mint sn = S[n - 1] * (mint(1) - POW[n]); FOR(i, n) { Y[i] = Y[i] * tPOW[n - 1 - i] * itPOW[n - 1] * iS[i] * iS[n - 1 - i]; if (i % 2 == 1) Y[i] = -Y[i]; } // sum_i Y[i] / 1-r^ix FOR(i, n) Y[i] *= itPOW[i]; vc<mint> f = middle_product(tPOW, Y); FOR(i, n) f[i] *= itPOW[i]; // prod 1-r^ix vc<mint> g(n); g[0] = mint(1); FOR(i, 1, n) { g[i] = tPOW[i] * sn * iS[i] * iS[n - i]; if (i % 2 == 1) g[i] = -g[i]; } f = convolution<mint>(f, g); f.resize(n); reverse(all(f)); mint ia = a.inverse(); mint pow = 1; FOR(i, n) f[i] *= pow, pow *= ia; return f; } #line 8 "test/3_yukicoder/2166.test.cpp" const int mod = 998244353; using mint = modint998; using poly = vc<mint>; using MAT = array<array<poly, 2>, 2>; struct Mono { using value_type = MAT; using X = value_type; static X op(X x, X y) { // これは Nlog^2N なので、雑で大丈夫 → そうでもない説 int nx = 0, ny = 0; FOR(i, 2) FOR(j, 2) chmax(nx, len(x[i][j])); FOR(i, 2) FOR(j, 2) chmax(ny, len(y[i][j])); int n = nx + ny - 1; int fft_len = 1; while (fft_len < n) fft_len *= 2; FOR(i, 2) FOR(j, 2) { x[i][j].resize(fft_len); ntt(x[i][j], false); } FOR(i, 2) FOR(j, 2) { y[i][j].resize(fft_len); ntt(y[i][j], false); } X z; FOR(i, 2) FOR(j, 2) z[i][j].resize(fft_len); FOR(i, 2) FOR(j, 2) FOR(k, 2) { FOR(p, fft_len) z[i][k][p] += x[i][j][p] * y[j][k][p]; } FOR(i, 2) FOR(j, 2) { ntt(z[i][j], true); z[i][j].resize(n); } return z; } static X unit() { MAT x; x[0][0] = x[1][1] = {mint(1)}; x[0][1] = x[1][0] = {}; return x; } static constexpr bool commute = 0; }; void solve_1(int Q) { VEC(pi, query, Q); auto make_mat = [&](ll K) -> MAT { MAT x; x[0][0] = {mint(-K - K), mint(2)}; // 2N-2K x[0][1] = {mint(-K * (K - 1) / 2), mint(K)}; // KN - K(K-1)/2 x[1][0] = {mint(1)}; x[1][1] = {}; return x; }; int MAX = 100'000; const int b_sz = 5010; const int b_num = ceil(MAX, b_sz) + 1; vvc<int> QID(b_num); FOR(q, Q) { auto [n, k] = query[q]; QID[k / b_sz].eb(q); } auto prod_range = [&](int L, int R) -> MAT { assert(L < R); vc<MAT> dat(R - L); FOR(i, R - L) dat[i] = make_mat(L + i); reverse(all(dat)); while (len(dat) > 1) { int n = len(dat); FOR(i, n) if (i % 2 == 1) { dat[i - 1] = Mono::op(dat[i - 1], dat[i]); } FOR(i, n) if (i % 2 == 0) dat[i / 2] = dat[i]; dat.resize(ceil(n, 2)); } return dat[0]; }; vc<mint> ANS(Q); MAT suffix_prod = Mono::unit(); FOR(b, b_num) { // suffix_prod に必要なものたちを ME する vc<mint> X; for (auto&& q: QID[b]) { X.eb(query[q].fi); } if (len(X)) { SubproductTree<mint> ST(X); auto Y0 = ST.evaluation(suffix_prod[0][0]); auto Y1 = ST.evaluation(suffix_prod[1][0]); FOR(t, len(X)) { int qid = QID[b][t]; auto [N, K] = query[qid]; N %= mod; pi p = {Y0[t].val, Y1[t].val}; FOR(k, b * b_sz, K) { ll c = k * (N + N - k + 1) / 2 % mod; p = {(2 * (N - k) * p.fi + c * p.se) % mod, p.fi}; } ANS[qid] = p.fi; } } suffix_prod = Mono::op(prod_range(b * b_sz, b * b_sz + b_sz), suffix_prod); } FOR(q, Q) print(ANS[q]); } mint solve_2(ll N, ll K) { if (K >= mod) return 0; assert(K <= mod); poly f = {mint(2 * N), mint(N)}; poly g = {mint(1), mint(2), inv<mint>(2)}; mint fa = [&]() -> mint { vc<mint> f = {1, 1}; return prefix_product_of_poly(f, K).val; }(); mint ANS = fa * from_log_differentiation_kth(K, f, g); return ANS; } void solve() { INT(T); if (T <= 10) { FOR(T) { LL(N, K); print(solve_2(N, K)); } return; } return solve_1(T); } signed main() { solve(); return 0; }