#line 1 "main.cpp" #define PROBLEM "https://yukicoder.me/problems/no/2206" #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; 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); } // (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 ceil(T x, U y) { return (x > 0 ? (x + y - 1) / y : x / y); } template T floor(T x, U y) { return (x > 0 ? x / y : (x - y + 1) / y); } template pair divmod(T x, U y) { T q = floor(x, y); return {q, x - q * y}; } template T SUM(const vector &A) { T sum = 0; for (auto &&a: A) sum += a; return sum; } #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) { assert(!que.empty()); T a = que.top(); que.pop(); return a; } template T POP(vc &que) { assert(!que.empty()); T a = que.back(); que.pop_back(); return a; } template ll binary_search(F check, ll ok, ll ng) { assert(check(ok)); while (abs(ok - ng) > 1) { auto x = (ng + ok) / 2; tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, x)); } return ok; } template double binary_search_real(F check, double ok, double ng, int iter = 100) { FOR(iter) { double x = (ok + ng) / 2; tie(ok, ng) = (check(x) ? mp(x, ng) : mp(ok, 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" // based on yosupo's fastio #include namespace fastio { #define FASTIO // クラスが read(), print() を持っているかを判定するメタ関数 struct has_write_impl { template static auto check(T &&x) -> decltype(x.write(), std::true_type{}); template static auto check(...) -> std::false_type; }; template class has_write : public decltype(has_write_impl::check(std::declval())) { }; struct has_read_impl { template static auto check(T &&x) -> decltype(x.read(), std::true_type{}); template static auto check(...) -> std::false_type; }; template class has_read : public decltype(has_read_impl::check(std::declval())) {}; struct Scanner { FILE *fp; char line[(1 << 15) + 1]; size_t st = 0, ed = 0; void reread() { memmove(line, line + st, ed - st); ed -= st; st = 0; ed += fread(line + ed, 1, (1 << 15) - ed, fp); line[ed] = '\0'; } bool succ() { while (true) { if (st == ed) { reread(); if (st == ed) return false; } while (st != ed && isspace(line[st])) st++; if (st != ed) break; } if (ed - st <= 50) { bool sep = false; for (size_t i = st; i < ed; i++) { if (isspace(line[i])) { sep = true; break; } } if (!sep) reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; while (true) { size_t sz = 0; while (st + sz < ed && !isspace(line[st + sz])) sz++; ref.append(line + st, sz); st += sz; if (!sz || st != ed) break; reread(); } return true; } template ::value, int> = 0> bool read_single(T &ref) { if (!succ()) return false; bool neg = false; if (line[st] == '-') { neg = true; st++; } ref = T(0); while (isdigit(line[st])) { ref = 10 * ref + (line[st++] & 0xf); } if (neg) ref = -ref; return true; } template ::value>::type * = nullptr> inline bool read_single(T &x) { x.read(); return true; } bool read_single(double &ref) { string s; if (!read_single(s)) return false; ref = std::stod(s); return true; } bool read_single(char &ref) { string s; if (!read_single(s) || s.size() != 1) return false; ref = s[0]; return true; } template bool read_single(vector &ref) { for (auto &d: ref) { if (!read_single(d)) return false; } return true; } template bool read_single(pair &p) { return (read_single(p.first) && read_single(p.second)); } template void read_single_tuple(T &t) { if constexpr (N < std::tuple_size::value) { auto &x = std::get(t); read_single(x); read_single_tuple(t); } } template bool read_single(tuple &tpl) { read_single_tuple(tpl); return true; } void read() {} template void read(H &h, T &... t) { bool f = read_single(h); assert(f); read(t...); } Scanner(FILE *fp) : fp(fp) {} }; struct Printer { Printer(FILE *_fp) : fp(_fp) {} ~Printer() { flush(); } static constexpr size_t SIZE = 1 << 15; FILE *fp; char line[SIZE], small[50]; size_t pos = 0; void flush() { fwrite(line, 1, pos, fp); pos = 0; } void write(const char val) { if (pos == SIZE) flush(); line[pos++] = val; } template ::value, int> = 0> void write(T val) { if (pos > (1 << 15) - 50) flush(); if (val == 0) { write('0'); return; } if (val < 0) { write('-'); val = -val; // todo min } size_t len = 0; while (val) { small[len++] = char(0x30 | (val % 10)); val /= 10; } for (size_t i = 0; i < len; i++) { line[pos + i] = small[len - 1 - i]; } pos += len; } void write(const string s) { for (char c: s) write(c); } void write(const char *s) { size_t len = strlen(s); for (size_t i = 0; i < len; i++) write(s[i]); } void write(const double x) { ostringstream oss; oss << fixed << setprecision(15) << x; string s = oss.str(); write(s); } void write(const long double x) { ostringstream oss; oss << fixed << setprecision(15) << x; string s = oss.str(); write(s); } template ::value>::type * = nullptr> inline void write(T x) { x.write(); } template void write(const vector val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } template void write(const pair val) { write(val.first); write(' '); write(val.second); } template void write_tuple(const T t) { if constexpr (N < std::tuple_size::value) { if constexpr (N > 0) { write(' '); } const auto x = std::get(t); write(x); write_tuple(t); } } template bool write(tuple tpl) { write_tuple(tpl); return true; } template void write(const array val) { auto n = val.size(); for (size_t i = 0; i < n; i++) { if (i) write(' '); write(val[i]); } } void write(i128 val) { string s; bool negative = 0; if (val < 0) { negative = 1; val = -val; } while (val) { s += '0' + int(val % 10); val /= 10; } if (negative) s += "-"; reverse(all(s)); if (len(s) == 0) s = "0"; write(s); } }; Scanner scanner = Scanner(stdin); Printer printer = Printer(stdout); void flush() { printer.flush(); } void print() { printer.write('\n'); } template void print(Head &&head, Tail &&... tail) { printer.write(head); if (sizeof...(Tail)) printer.write(' '); print(forward(tail)...); } void read() {} template void read(Head &head, Tail &... tail) { scanner.read(head); read(tail...); } } // namespace fastio using fastio::print; using fastio::flush; using fastio::read; #define INT(...) \ int __VA_ARGS__; \ read(__VA_ARGS__) #define LL(...) \ ll __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 2 "/home/maspy/compro/library/mod/modint.hpp" template struct modint { int val; constexpr modint(ll x = 0) noexcept { if (0 <= x && x < mod) val = x; else { x %= mod; val = (x < 0 ? x + mod : x); } } bool operator<(const modint &other) const { return val < other.val; } // To use std::map modint &operator+=(const modint &p) { if ((val += p.val) >= mod) val -= mod; return *this; } modint &operator-=(const modint &p) { if ((val += mod - p.val) >= mod) val -= mod; return *this; } modint &operator*=(const modint &p) { val = (int)(1LL * val * p.val % mod); return *this; } modint &operator/=(const modint &p) { *this *= p.inverse(); return *this; } modint operator-() const { return modint(-val); } 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(int64_t n) const { modint ret(1), mul(val); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } #ifdef FASTIO void write() { fastio::printer.write(val); } void read() { ll x; fastio::scanner.read(x); if (x < 0 || x >= mod) x %= mod; if (x < 0) x += mod; val += x; } #endif static constexpr int get_mod() { return mod; } }; struct ArbitraryModInt { static constexpr bool is_modint = true; int val; ArbitraryModInt() : val(0) {} ArbitraryModInt(int64_t y) : val(y >= 0 ? y % get_mod() : (get_mod() - (-y) % get_mod()) % get_mod()) {} bool operator<(const ArbitraryModInt &other) const { return val < other.val; } // To use std::map static int &get_mod() { static int mod = 0; return mod; } static void set_mod(int md) { get_mod() = md; } ArbitraryModInt &operator+=(const ArbitraryModInt &p) { if ((val += p.val) >= get_mod()) val -= get_mod(); return *this; } ArbitraryModInt &operator-=(const ArbitraryModInt &p) { if ((val += get_mod() - p.val) >= get_mod()) val -= get_mod(); return *this; } ArbitraryModInt &operator*=(const ArbitraryModInt &p) { long long a = (long long)val * p.val; int xh = (int)(a >> 32), xl = (int)a, d, m; asm("divl %4; \n\t" : "=a"(d), "=d"(m) : "d"(xh), "a"(xl), "r"(get_mod())); val = m; return *this; } ArbitraryModInt &operator/=(const ArbitraryModInt &p) { *this *= p.inverse(); return *this; } ArbitraryModInt operator-() const { return ArbitraryModInt(get_mod() - val); } ArbitraryModInt operator+(const ArbitraryModInt &p) const { return ArbitraryModInt(*this) += p; } ArbitraryModInt operator-(const ArbitraryModInt &p) const { return ArbitraryModInt(*this) -= p; } ArbitraryModInt operator*(const ArbitraryModInt &p) const { return ArbitraryModInt(*this) *= p; } ArbitraryModInt operator/(const ArbitraryModInt &p) const { return ArbitraryModInt(*this) /= p; } bool operator==(const ArbitraryModInt &p) const { return val == p.val; } bool operator!=(const ArbitraryModInt &p) const { return val != p.val; } ArbitraryModInt inverse() const { int a = val, b = get_mod(), u = 1, v = 0, t; while (b > 0) { t = a / b; swap(a -= t * b, b), swap(u -= t * v, v); } return ArbitraryModInt(u); } ArbitraryModInt pow(int64_t n) const { ArbitraryModInt ret(1), mul(val); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } #ifdef FASTIO void write() { fastio::printer.write(val); } void read() { fastio::scanner.read(val); } #endif }; 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 (int(dat.size()) <= n) { int k = dat.size(); auto q = (mod + k - 1) / k; int r = k * q - mod; dat.emplace_back(dat[r] * mint(q)); } return dat[n]; } template mint fact(int n) { static const int mod = mint::get_mod(); static vector dat = {1, 1}; assert(0 <= n); if (n >= mod) return 0; while (int(dat.size()) <= n) { int k = dat.size(); dat.emplace_back(dat[k - 1] * mint(k)); } return dat[n]; } template mint fact_inv(int n) { static const int mod = mint::get_mod(); static vector dat = {1, 1}; assert(-1 <= n && n < mod); if (n == -1) return mint(0); while (int(dat.size()) <= n) { int k = dat.size(); dat.emplace_back(dat[k - 1] * inv(k)); } 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 (dense) return C_dense(n, k); if (!large) return fact(n) * fact_inv(k) * fact_inv(n - k); k = min(k, n - k); mint x(1); FOR(i, k) { x *= mint(n - i); } x *= fact_inv(k); return x; } 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} の計算 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); } using modint107 = modint<1000000007>; using modint998 = modint<998244353>; using amint = ArbitraryModInt; #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 1 "/home/maspy/compro/library/mod/prefix_sum_of_binom.hpp" // \sum_{k=0}^m binom(n,k) を求めるクエリにオンラインで答える template struct Prefix_Sum_Of_Binom { const int MAX_N; const int B; vvc dat; Prefix_Sum_Of_Binom(int MAX_N) : MAX_N(MAX_N), B(sqrt(MAX_N + 1)) { int n = MAX_N / B; dat.assign(n + 1, vc(MAX_N + 1)); FOR(k, 1, n + 1) { FOR(i, k * B, MAX_N) { dat[k][i + 1] = dat[k][i] + dat[k][i] - C(i, k * B - 1); } } } // \sum_{k=0}^m binom(n,k) mint query(int n, int m) { if (m == 0) return mint(0); assert(n <= MAX_N); int k = m / B; mint x = dat[k][n]; FOR(i, k * B, m) x += C(n, i); return x; } }; #line 8 "main.cpp" using mint = modint998; void solve() { LL(Q); const int LIM = 200'000; vc POW = powertable_1(2, LIM + 10); Prefix_Sum_Of_Binom X(LIM); FOR(Q) { INT(n, m); mint cf = POW[n] - mint(1); mint x = X.query(n - 1, m); print(cf * x); } } signed main() { solve(); return 0; }