/** * date : 2023-10-20 22:19:20 * author : Nyaan */ #define NDEBUG using namespace std; // intrinstic #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include // utility namespace Nyaan { using ll = long long; using i64 = long long; using u64 = unsigned long long; using i128 = __int128_t; using u128 = __uint128_t; template using V = vector; template using VV = vector>; using vi = vector; using vl = vector; using vd = V; using vs = V; using vvi = vector>; using vvl = vector>; template using minpq = priority_queue, greater>; template struct P : pair { template P(Args... args) : pair(args...) {} using pair::first; using pair::second; P &operator+=(const P &r) { first += r.first; second += r.second; return *this; } P &operator-=(const P &r) { first -= r.first; second -= r.second; return *this; } P &operator*=(const P &r) { first *= r.first; second *= r.second; return *this; } template P &operator*=(const S &r) { first *= r, second *= r; return *this; } P operator+(const P &r) const { return P(*this) += r; } P operator-(const P &r) const { return P(*this) -= r; } P operator*(const P &r) const { return P(*this) *= r; } template P operator*(const S &r) const { return P(*this) *= r; } P operator-() const { return P{-first, -second}; } }; using pl = P; using pi = P; using vp = V; constexpr int inf = 1001001001; constexpr long long infLL = 4004004004004004004LL; template int sz(const T &t) { return t.size(); } template inline bool amin(T &x, U y) { return (y < x) ? (x = y, true) : false; } template inline bool amax(T &x, U y) { return (x < y) ? (x = y, true) : false; } template inline T Max(const vector &v) { return *max_element(begin(v), end(v)); } template inline T Min(const vector &v) { return *min_element(begin(v), end(v)); } template inline long long Sum(const vector &v) { return accumulate(begin(v), end(v), 0LL); } template int lb(const vector &v, const T &a) { return lower_bound(begin(v), end(v), a) - begin(v); } template int ub(const vector &v, const T &a) { return upper_bound(begin(v), end(v), a) - begin(v); } constexpr long long TEN(int n) { long long ret = 1, x = 10; for (; n; x *= x, n >>= 1) ret *= (n & 1 ? x : 1); return ret; } template pair mkp(const T &t, const U &u) { return make_pair(t, u); } template vector mkrui(const vector &v, bool rev = false) { vector ret(v.size() + 1); if (rev) { for (int i = int(v.size()) - 1; i >= 0; i--) ret[i] = v[i] + ret[i + 1]; } else { for (int i = 0; i < int(v.size()); i++) ret[i + 1] = ret[i] + v[i]; } return ret; }; template vector mkuni(const vector &v) { vector ret(v); sort(ret.begin(), ret.end()); ret.erase(unique(ret.begin(), ret.end()), ret.end()); return ret; } template vector mkord(int N, F f) { vector ord(N); iota(begin(ord), end(ord), 0); sort(begin(ord), end(ord), f); return ord; } template vector mkinv(vector &v) { int max_val = *max_element(begin(v), end(v)); vector inv(max_val + 1, -1); for (int i = 0; i < (int)v.size(); i++) inv[v[i]] = i; return inv; } vector mkiota(int n) { vector ret(n); iota(begin(ret), end(ret), 0); return ret; } template T mkrev(const T &v) { T w{v}; reverse(begin(w), end(w)); return w; } template bool nxp(vector &v) { return next_permutation(begin(v), end(v)); } // 返り値の型は入力の T に依存 // i 要素目 : [0, a[i]) template vector> product(const vector &a) { vector> ret; vector v; auto dfs = [&](auto rc, int i) -> void { if (i == (int)a.size()) { ret.push_back(v); return; } for (int j = 0; j < a[i]; j++) v.push_back(j), rc(rc, i + 1), v.pop_back(); }; dfs(dfs, 0); return ret; } // F : function(void(T&)), mod を取る操作 // T : 整数型のときはオーバーフローに注意する template T Power(T a, long long n, const T &I, const function &f) { T res = I; for (; n; f(a = a * a), n >>= 1) { if (n & 1) f(res = res * a); } return res; } // T : 整数型のときはオーバーフローに注意する template T Power(T a, long long n, const T &I = T{1}) { return Power(a, n, I, function{[](T &) -> void {}}); } template T Rev(const T &v) { T res = v; reverse(begin(res), end(res)); return res; } template vector Transpose(const vector &v) { using U = typename T::value_type; int H = v.size(), W = v[0].size(); vector res(W, T(H, U{})); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { res[j][i] = v[i][j]; } } return res; } template vector Rotate(const vector &v, int clockwise = true) { using U = typename T::value_type; int H = v.size(), W = v[0].size(); vector res(W, T(H, U{})); for (int i = 0; i < H; i++) { for (int j = 0; j < W; j++) { if (clockwise) { res[W - 1 - j][i] = v[i][j]; } else { res[j][H - 1 - i] = v[i][j]; } } } return res; } } // namespace Nyaan // bit operation namespace Nyaan { __attribute__((target("popcnt"))) inline int popcnt(const u64 &a) { return _mm_popcnt_u64(a); } inline int lsb(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int ctz(const u64 &a) { return a ? __builtin_ctzll(a) : 64; } inline int msb(const u64 &a) { return a ? 63 - __builtin_clzll(a) : -1; } template inline int gbit(const T &a, int i) { return (a >> i) & 1; } template inline void sbit(T &a, int i, bool b) { if (gbit(a, i) != b) a ^= T(1) << i; } constexpr long long PW(int n) { return 1LL << n; } constexpr long long MSK(int n) { return (1LL << n) - 1; } } // namespace Nyaan // inout namespace Nyaan { template ostream &operator<<(ostream &os, const pair &p) { os << p.first << " " << p.second; return os; } template istream &operator>>(istream &is, pair &p) { is >> p.first >> p.second; return is; } template ostream &operator<<(ostream &os, const vector &v) { int s = (int)v.size(); for (int i = 0; i < s; i++) os << (i ? " " : "") << v[i]; return os; } template istream &operator>>(istream &is, vector &v) { for (auto &x : v) is >> x; return is; } istream &operator>>(istream &is, __int128_t &x) { string S; is >> S; x = 0; int flag = 0; for (auto &c : S) { if (c == '-') { flag = true; continue; } x *= 10; x += c - '0'; } if (flag) x = -x; return is; } istream &operator>>(istream &is, __uint128_t &x) { string S; is >> S; x = 0; for (auto &c : S) { x *= 10; x += c - '0'; } return is; } ostream &operator<<(ostream &os, __int128_t x) { if (x == 0) return os << 0; if (x < 0) os << '-', x = -x; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } ostream &operator<<(ostream &os, __uint128_t x) { if (x == 0) return os << 0; string S; while (x) S.push_back('0' + x % 10), x /= 10; reverse(begin(S), end(S)); return os << S; } void in() {} template void in(T &t, U &...u) { cin >> t; in(u...); } void out() { cout << "\n"; } template void out(const T &t, const U &...u) { cout << t; if (sizeof...(u)) cout << sep; out(u...); } struct IoSetupNya { IoSetupNya() { cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(15); cerr << fixed << setprecision(7); } } iosetupnya; } // namespace Nyaan // debug #ifdef NyaanDebug #define trc(...) (void(0)) #else #define trc(...) (void(0)) #endif #ifdef NyaanLocal #define trc2(...) (void(0)) #else #define trc2(...) (void(0)) #endif // macro #define each(x, v) for (auto&& x : v) #define each2(x, y, v) for (auto&& [x, y] : v) #define all(v) (v).begin(), (v).end() #define rep(i, N) for (long long i = 0; i < (long long)(N); i++) #define repr(i, N) for (long long i = (long long)(N)-1; i >= 0; i--) #define rep1(i, N) for (long long i = 1; i <= (long long)(N); i++) #define repr1(i, N) for (long long i = (N); (long long)(i) > 0; i--) #define reg(i, a, b) for (long long i = (a); i < (b); i++) #define regr(i, a, b) for (long long i = (b)-1; i >= (a); i--) #define fi first #define se second #define ini(...) \ int __VA_ARGS__; \ in(__VA_ARGS__) #define inl(...) \ long long __VA_ARGS__; \ in(__VA_ARGS__) #define ins(...) \ string __VA_ARGS__; \ in(__VA_ARGS__) #define in2(s, t) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i]); \ } #define in3(s, t, u) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i]); \ } #define in4(s, t, u, v) \ for (int i = 0; i < (int)s.size(); i++) { \ in(s[i], t[i], u[i], v[i]); \ } #define die(...) \ do { \ Nyaan::out(__VA_ARGS__); \ return; \ } while (0) namespace Nyaan { void solve(); } int main() { Nyaan::solve(); } // using namespace std; // {rank, det(非正方行列の場合は未定義)} を返す // 型が double や Rational でも動くはず?(未検証) // // pivot 候補 : [0, pivot_end) template std::pair GaussElimination(vector> &a, int pivot_end = -1, bool diagonalize = false) { int H = a.size(), W = a[0].size(), rank = 0; if (pivot_end == -1) pivot_end = W; T det = 1; for (int j = 0; j < pivot_end; j++) { int idx = -1; for (int i = rank; i < H; i++) { if (a[i][j] != T(0)) { idx = i; break; } } if (idx == -1) { det = 0; continue; } if (rank != idx) det = -det, swap(a[rank], a[idx]); det *= a[rank][j]; if (diagonalize && a[rank][j] != T(1)) { T coeff = T(1) / a[rank][j]; for (int k = j; k < W; k++) a[rank][k] *= coeff; } int is = diagonalize ? 0 : rank + 1; for (int i = is; i < H; i++) { if (i == rank) continue; if (a[i][j] != T(0)) { T coeff = a[i][j] / a[rank][j]; for (int k = j; k < W; k++) a[i][k] -= a[rank][k] * coeff; } } rank++; } return make_pair(rank, det); } // 解が存在する場合は, 解が v + C_1 w_1 + ... + C_k w_k と表せるとして // (v, w_1, ..., w_k) を返す // 解が存在しない場合は空のベクトルを返す // // double や Rational でも動くはず?(未検証) template vector> LinearEquation(vector> a, vector b) { int H = a.size(), W = a[0].size(); for (int i = 0; i < H; i++) a[i].push_back(b[i]); auto p = GaussElimination(a, W, true); int rank = p.first; for (int i = rank; i < H; ++i) { if (a[i][W] != 0) return vector>{}; } vector> res(1, vector(W)); vector pivot(W, -1); for (int i = 0, j = 0; i < rank; ++i) { while (a[i][j] == 0) ++j; res[0][j] = a[i][W], pivot[j] = i; } for (int j = 0; j < W; ++j) { if (pivot[j] == -1) { vector x(W); x[j] = 1; for (int k = 0; k < j; ++k) { if (pivot[k] != -1) x[k] = -a[pivot[k]][j]; } res.push_back(x); } } return res; } template struct FormalPowerSeries : vector { using vector::vector; using FPS = FormalPowerSeries; FPS &operator+=(const FPS &r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i]; return *this; } FPS &operator+=(const mint &r) { if (this->empty()) this->resize(1); (*this)[0] += r; return *this; } FPS &operator-=(const FPS &r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i]; return *this; } FPS &operator-=(const mint &r) { if (this->empty()) this->resize(1); (*this)[0] -= r; return *this; } FPS &operator*=(const mint &v) { for (int k = 0; k < (int)this->size(); k++) (*this)[k] *= v; return *this; } FPS &operator/=(const FPS &r) { if (this->size() < r.size()) { this->clear(); return *this; } int n = this->size() - r.size() + 1; if ((int)r.size() <= 64) { FPS f(*this), g(r); g.shrink(); mint coeff = g.back().inverse(); for (auto &x : g) x *= coeff; int deg = (int)f.size() - (int)g.size() + 1; int gs = g.size(); FPS quo(deg); for (int i = deg - 1; i >= 0; i--) { quo[i] = f[i + gs - 1]; for (int j = 0; j < gs; j++) f[i + j] -= quo[i] * g[j]; } *this = quo * coeff; this->resize(n, mint(0)); return *this; } return *this = ((*this).rev().pre(n) * r.rev().inv(n)).pre(n).rev(); } FPS &operator%=(const FPS &r) { *this -= *this / r * r; shrink(); return *this; } FPS operator+(const FPS &r) const { return FPS(*this) += r; } FPS operator+(const mint &v) const { return FPS(*this) += v; } FPS operator-(const FPS &r) const { return FPS(*this) -= r; } FPS operator-(const mint &v) const { return FPS(*this) -= v; } FPS operator*(const FPS &r) const { return FPS(*this) *= r; } FPS operator*(const mint &v) const { return FPS(*this) *= v; } FPS operator/(const FPS &r) const { return FPS(*this) /= r; } FPS operator%(const FPS &r) const { return FPS(*this) %= r; } FPS operator-() const { FPS ret(this->size()); for (int i = 0; i < (int)this->size(); i++) ret[i] = -(*this)[i]; return ret; } void shrink() { while (this->size() && this->back() == mint(0)) this->pop_back(); } FPS rev() const { FPS ret(*this); reverse(begin(ret), end(ret)); return ret; } FPS dot(FPS r) const { FPS ret(min(this->size(), r.size())); for (int i = 0; i < (int)ret.size(); i++) ret[i] = (*this)[i] * r[i]; return ret; } // 前 sz 項を取ってくる。sz に足りない項は 0 埋めする FPS pre(int sz) const { FPS ret(begin(*this), begin(*this) + min((int)this->size(), sz)); if ((int)ret.size() < sz) ret.resize(sz); return ret; } FPS operator>>(int sz) const { if ((int)this->size() <= sz) return {}; FPS ret(*this); ret.erase(ret.begin(), ret.begin() + sz); return ret; } FPS operator<<(int sz) const { FPS ret(*this); ret.insert(ret.begin(), sz, mint(0)); return ret; } FPS diff() const { const int n = (int)this->size(); FPS ret(max(0, n - 1)); mint one(1), coeff(1); for (int i = 1; i < n; i++) { ret[i - 1] = (*this)[i] * coeff; coeff += one; } return ret; } FPS integral() const { const int n = (int)this->size(); FPS ret(n + 1); ret[0] = mint(0); if (n > 0) ret[1] = mint(1); auto mod = mint::get_mod(); for (int i = 2; i <= n; i++) ret[i] = (-ret[mod % i]) * (mod / i); for (int i = 0; i < n; i++) ret[i + 1] *= (*this)[i]; return ret; } mint eval(mint x) const { mint r = 0, w = 1; for (auto &v : *this) r += w * v, w *= x; return r; } FPS log(int deg = -1) const { assert(!(*this).empty() && (*this)[0] == mint(1)); if (deg == -1) deg = (int)this->size(); return (this->diff() * this->inv(deg)).pre(deg - 1).integral(); } FPS pow(int64_t k, int deg = -1) const { const int n = (int)this->size(); if (deg == -1) deg = n; if (k == 0) { FPS ret(deg); if (deg) ret[0] = 1; return ret; } for (int i = 0; i < n; i++) { if ((*this)[i] != mint(0)) { mint rev = mint(1) / (*this)[i]; FPS ret = (((*this * rev) >> i).log(deg) * k).exp(deg); ret *= (*this)[i].pow(k); ret = (ret << (i * k)).pre(deg); if ((int)ret.size() < deg) ret.resize(deg, mint(0)); return ret; } if (__int128_t(i + 1) * k >= deg) return FPS(deg, mint(0)); } return FPS(deg, mint(0)); } static void *ntt_ptr; static void set_fft(); FPS &operator*=(const FPS &r); void ntt(); void intt(); void ntt_doubling(); static int ntt_pr(); FPS inv(int deg = -1) const; FPS exp(int deg = -1) const; }; template void *FormalPowerSeries::ntt_ptr = nullptr; /** * @brief 多項式/形式的冪級数ライブラリ * @docs docs/fps/formal-power-series.md */ using namespace std; // コンストラクタの MAX に 「C(n, r) や fac(n) でクエリを投げる最大の n 」 // を入れると倍速くらいになる // mod を超えて前計算して 0 割りを踏むバグは対策済み template struct Binomial { vector f, g, h; Binomial(int MAX = 0) { assert(T::get_mod() != 0 && "Binomial()"); f.resize(1, T{1}); g.resize(1, T{1}); h.resize(1, T{1}); if (MAX > 0) extend(MAX + 1); } void extend(int m = -1) { int n = f.size(); if (m == -1) m = n * 2; m = min(m, T::get_mod()); if (n >= m) return; f.resize(m); g.resize(m); h.resize(m); for (int i = n; i < m; i++) f[i] = f[i - 1] * T(i); g[m - 1] = f[m - 1].inverse(); h[m - 1] = g[m - 1] * f[m - 2]; for (int i = m - 2; i >= n; i--) { g[i] = g[i + 1] * T(i + 1); h[i] = g[i] * f[i - 1]; } } T fac(int i) { if (i < 0) return T(0); while (i >= (int)f.size()) extend(); return f[i]; } T finv(int i) { if (i < 0) return T(0); while (i >= (int)g.size()) extend(); return g[i]; } T inv(int i) { if (i < 0) return -inv(-i); while (i >= (int)h.size()) extend(); return h[i]; } T C(int n, int r) { if (n < 0 || n < r || r < 0) return T(0); return fac(n) * finv(n - r) * finv(r); } inline T operator()(int n, int r) { return C(n, r); } template T multinomial(const vector& r) { static_assert(is_integral::value == true); int n = 0; for (auto& x : r) { if (x < 0) return T(0); n += x; } T res = fac(n); for (auto& x : r) res *= finv(x); return res; } template T operator()(const vector& r) { return multinomial(r); } T C_naive(int n, int r) { if (n < 0 || n < r || r < 0) return T(0); T ret = T(1); r = min(r, n - r); for (int i = 1; i <= r; ++i) ret *= inv(i) * (n--); return ret; } T P(int n, int r) { if (n < 0 || n < r || r < 0) return T(0); return fac(n) * finv(n - r); } // [x^r] 1 / (1-x)^n T H(int n, int r) { if (n < 0 || r < 0) return T(0); return r == 0 ? 1 : C(n + r - 1, r); } }; // input : y(0), y(1), ..., y(n - 1) // output : y(t), y(t + 1), ..., y(t + m - 1) // (if m is default, m = n) template FormalPowerSeries SamplePointShift(FormalPowerSeries& y, mint t, int m = -1) { if (m == -1) m = y.size(); long long T = t.get(); int k = (int)y.size() - 1; T %= mint::get_mod(); if (T <= k) { FormalPowerSeries ret(m); int ptr = 0; for (int64_t i = T; i <= k and ptr < m; i++) { ret[ptr++] = y[i]; } if (k + 1 < T + m) { auto suf = SamplePointShift(y, k + 1, m - ptr); for (int i = k + 1; i < T + m; i++) { ret[ptr++] = suf[i - (k + 1)]; } } return ret; } if (T + m > mint::get_mod()) { auto pref = SamplePointShift(y, T, mint::get_mod() - T); auto suf = SamplePointShift(y, 0, m - pref.size()); copy(begin(suf), end(suf), back_inserter(pref)); return pref; } FormalPowerSeries finv(k + 1, 1), d(k + 1); for (int i = 2; i <= k; i++) finv[k] *= i; finv[k] = mint(1) / finv[k]; for (int i = k; i >= 1; i--) finv[i - 1] = finv[i] * i; for (int i = 0; i <= k; i++) { d[i] = finv[i] * finv[k - i] * y[i]; if ((k - i) & 1) d[i] = -d[i]; } FormalPowerSeries h(m + k); for (int i = 0; i < m + k; i++) { h[i] = mint(1) / (T - k + i); } auto dh = d * h; FormalPowerSeries ret(m); mint cur = T; for (int i = 1; i <= k; i++) cur *= T - i; for (int i = 0; i < m; i++) { ret[i] = cur * dh[k + i]; cur *= T + i + 1; cur *= h[i]; } return ret; } template vector> inverse_matrix(const vector>& a) { int N = a.size(); assert(N > 0); assert(N == (int)a[0].size()); vector> m(N, vector(2 * N)); for (int i = 0; i < N; i++) { copy(begin(a[i]), end(a[i]), begin(m[i])); m[i][N + i] = 1; } auto [rank, det] = GaussElimination(m, N, true); if (rank != N) return {}; vector> b(N); for (int i = 0; i < N; i++) { copy(begin(m[i]) + N, end(m[i]), back_inserter(b[i])); } return b; } template struct Matrix { vector > A; Matrix() = default; Matrix(int n, int m) : A(n, vector(m, T())) {} Matrix(int n) : A(n, vector(n, T())){}; int H() const { return A.size(); } int W() const { return A[0].size(); } int size() const { return A.size(); } inline const vector &operator[](int k) const { return A[k]; } inline vector &operator[](int k) { return A[k]; } static Matrix I(int n) { Matrix mat(n); for (int i = 0; i < n; i++) mat[i][i] = 1; return (mat); } Matrix &operator+=(const Matrix &B) { int n = H(), m = W(); assert(n == B.H() && m == B.W()); for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) (*this)[i][j] += B[i][j]; return (*this); } Matrix &operator-=(const Matrix &B) { int n = H(), m = W(); assert(n == B.H() && m == B.W()); for (int i = 0; i < n; i++) for (int j = 0; j < m; j++) (*this)[i][j] -= B[i][j]; return (*this); } Matrix &operator*=(const Matrix &B) { int n = H(), m = B.W(), p = W(); assert(p == B.H()); vector > C(n, vector(m, T{})); for (int i = 0; i < n; i++) for (int k = 0; k < p; k++) for (int j = 0; j < m; j++) C[i][j] += (*this)[i][k] * B[k][j]; A.swap(C); return (*this); } Matrix &operator^=(long long k) { Matrix B = Matrix::I(H()); while (k > 0) { if (k & 1) B *= *this; *this *= *this; k >>= 1LL; } A.swap(B.A); return (*this); } Matrix operator+(const Matrix &B) const { return (Matrix(*this) += B); } Matrix operator-(const Matrix &B) const { return (Matrix(*this) -= B); } Matrix operator*(const Matrix &B) const { return (Matrix(*this) *= B); } Matrix operator^(const long long k) const { return (Matrix(*this) ^= k); } bool operator==(const Matrix &B) const { assert(H() == B.H() && W() == B.W()); for (int i = 0; i < H(); i++) for (int j = 0; j < W(); j++) if (A[i][j] != B[i][j]) return false; return true; } bool operator!=(const Matrix &B) const { assert(H() == B.H() && W() == B.W()); for (int i = 0; i < H(); i++) for (int j = 0; j < W(); j++) if (A[i][j] != B[i][j]) return true; return false; } Matrix inverse() const { assert(H() == W()); Matrix B(H()); B.A = inverse_matrix(A); return B; } friend ostream &operator<<(ostream &os, const Matrix &p) { int n = p.H(), m = p.W(); for (int i = 0; i < n; i++) { os << (i ? " " : "") << "["; for (int j = 0; j < m; j++) { os << p[i][j] << (j + 1 == m ? "]\n" : ","); } } return (os); } T determinant() const { Matrix B(*this); assert(H() == W()); T ret = 1; for (int i = 0; i < H(); i++) { int idx = -1; for (int j = i; j < W(); j++) { if (B[j][i] != 0) { idx = j; break; } } if (idx == -1) return 0; if (i != idx) { ret *= T(-1); swap(B[i], B[idx]); } ret *= B[i][i]; T inv = T(1) / B[i][i]; for (int j = 0; j < W(); j++) { B[i][j] *= inv; } for (int j = i + 1; j < H(); j++) { T a = B[j][i]; if (a == 0) continue; for (int k = i; k < W(); k++) { B[j][k] -= B[i][k] * a; } } } return ret; } }; /** * @brief 行列ライブラリ */ // return m(k-1) * m(k-2) * ... * m(1) * m(0) template Matrix polynomial_matrix_prod(Matrix> &m, long long k) { using Mat = Matrix; using fps = FormalPowerSeries; auto shift = [](vector &G, mint x) -> vector { int d = G.size(), n = G[0].size(); vector H(d, Mat(n)); for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { fps g(d); for (int l = 0; l < d; l++) g[l] = G[l][i][j]; fps h = SamplePointShift(g, x); for (int l = 0; l < d; l++) H[l][i][j] = h[l]; } } return H; }; int n = m.size(); int deg = 1; for (auto &_ : m.A) { for (auto &x : _) deg = max(deg, (int)x.size() - 1); } while (deg & (deg - 1)) deg++; vector G(deg + 1); long long v = 1; while (deg * v * v < k) v *= 2; mint iv = mint(v).inverse(); for (int i = 0; i < (int)G.size(); i++) { mint x = mint(v) * i; Mat mt(n); for (int j = 0; j < n; j++) for (int l = 0; l < n; l++) mt[j][l] = m[j][l].eval(x); G[i] = mt; } for (long long w = 1; w != v; w <<= 1) { mint W = w; auto G1 = shift(G, W * iv); auto G2 = shift(G, (W * deg * v + v) * iv); auto G3 = shift(G, (W * deg * v + v + W) * iv); for (int i = 0; i <= w * deg; i++) G[i] = G1[i] * G[i], G2[i] = G3[i] * G2[i]; copy(begin(G2), end(G2) - 1, back_inserter(G)); } Mat res = Mat::I(n); long long i = 0; while (i + v <= k) res = G[i / v] * res, i += v; while (i < k) { Mat mt(n); for (int j = 0; j < n; j++) for (int l = 0; l < n; l++) mt[j][l] = m[j][l].eval(i); res = mt * res; i++; } return res; } /** * @brief 多項式行列のprefix product */ // return polynomial coefficient s.t. sum_{j=k...0} f_j(i) a_{i+j} = 0 // (In more details, read verification code.) template vector> find_p_recursive(vector& a, int d) { using fps = FormalPowerSeries; int n = a.size(); int k = (n + 2) / (d + 2) - 1; if (k <= 0) return {}; int m = (k + 1) * (d + 1); vector> M(m - 1, vector(m)); for (int i = 0; i < m - 1; i++) { for (int j = 0; j <= k; j++) { mint base = 1; for (int l = 0; l <= d; l++) { M[i][(d + 1) * j + l] = base * a[i + j]; base *= i + j; } } } auto gauss = LinearEquation(M, vector(m - 1, 0)); if (gauss.size() <= 1) return {}; auto c = gauss[1]; while (all_of(end(c) - d - 1, end(c), [](mint x) { return x == mint(0); })) { c.erase(end(c) - d - 1, end(c)); } k = c.size() / (d + 1) - 1; vector res; for (int i = 0, j = 0; i < (int)c.size(); i += d + 1, j++) { fps f{1}, base{j, 1}; fps sm; for (int l = 0; l <= d; l++) sm += f * c[i + l], f *= base; res.push_back(sm); } reverse(begin(res), end(res)); return res; } template mint kth_term_of_p_recursive(vector& a, long long k, int d) { if (k < (int)a.size()) return a[k]; if (all_of(begin(a), end(a), [](mint x) { return x == mint(0); })) return 0; auto fs = find_p_recursive(a, d); assert(fs.empty() == false); int deg = fs.size() - 1; assert(deg >= 1); Matrix> m(deg), denom(1); for (int i = 0; i < deg; i++) m[0][i] = -fs[i + 1]; for (int i = 1; i < deg; i++) m[i][i - 1] = fs[0]; denom[0][0] = fs[0]; Matrix a0(deg); for (int i = 0; i < deg; i++) a0[i][0] = a[deg - 1 - i]; mint res = (polynomial_matrix_prod(m, k - deg + 1) * a0)[0][0]; res /= polynomial_matrix_prod(denom, k - deg + 1)[0][0]; return res; } template mint kth_term_of_p_recursive(vector& a, long long k) { if (k < (int)a.size()) return a[k]; if (all_of(begin(a), end(a), [](mint x) { return x == mint(0); })) return 0; int n = a.size() - 1; vector b{begin(a), end(a) - 1}; for (int d = 0;; d++) { #ifdef NyaanLocal cerr << "d = " << d << endl; #endif if ((n + 2) / (d + 2) <= 1) break; if (kth_term_of_p_recursive(b, n, d) == a.back()) { #ifdef NyaanLocal cerr << "Found, d = " << d << endl; #endif return kth_term_of_p_recursive(a, k, d); } } cerr << "Failed." << endl; exit(1); } /** * @brief P-recursiveの高速計算 * @docs docs/fps/find-p-recursive.md */ // template struct NTT { static constexpr uint32_t get_pr() { uint32_t _mod = mint::get_mod(); using u64 = uint64_t; u64 ds[32] = {}; int idx = 0; u64 m = _mod - 1; for (u64 i = 2; i * i <= m; ++i) { if (m % i == 0) { ds[idx++] = i; while (m % i == 0) m /= i; } } if (m != 1) ds[idx++] = m; uint32_t _pr = 2; while (1) { int flg = 1; for (int i = 0; i < idx; ++i) { u64 a = _pr, b = (_mod - 1) / ds[i], r = 1; while (b) { if (b & 1) r = r * a % _mod; a = a * a % _mod; b >>= 1; } if (r == 1) { flg = 0; break; } } if (flg == 1) break; ++_pr; } return _pr; }; static constexpr uint32_t mod = mint::get_mod(); static constexpr uint32_t pr = get_pr(); static constexpr int level = __builtin_ctzll(mod - 1); mint dw[level], dy[level]; void setwy(int k) { mint w[level], y[level]; w[k - 1] = mint(pr).pow((mod - 1) / (1 << k)); y[k - 1] = w[k - 1].inverse(); for (int i = k - 2; i > 0; --i) w[i] = w[i + 1] * w[i + 1], y[i] = y[i + 1] * y[i + 1]; dw[1] = w[1], dy[1] = y[1], dw[2] = w[2], dy[2] = y[2]; for (int i = 3; i < k; ++i) { dw[i] = dw[i - 1] * y[i - 2] * w[i]; dy[i] = dy[i - 1] * w[i - 2] * y[i]; } } NTT() { setwy(level); } void fft4(vector &a, int k) { if ((int)a.size() <= 1) return; if (k == 1) { mint a1 = a[1]; a[1] = a[0] - a[1]; a[0] = a[0] + a1; return; } if (k & 1) { int v = 1 << (k - 1); for (int j = 0; j < v; ++j) { mint ajv = a[j + v]; a[j + v] = a[j] - ajv; a[j] += ajv; } } int u = 1 << (2 + (k & 1)); int v = 1 << (k - 2 - (k & 1)); mint one = mint(1); mint imag = dw[1]; while (v) { // jh = 0 { int j0 = 0; int j1 = v; int j2 = j1 + v; int j3 = j2 + v; for (; j0 < v; ++j0, ++j1, ++j2, ++j3) { mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3]; mint t0p2 = t0 + t2, t1p3 = t1 + t3; mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag; a[j0] = t0p2 + t1p3, a[j1] = t0p2 - t1p3; a[j2] = t0m2 + t1m3, a[j3] = t0m2 - t1m3; } } // jh >= 1 mint ww = one, xx = one * dw[2], wx = one; for (int jh = 4; jh < u;) { ww = xx * xx, wx = ww * xx; int j0 = jh * v; int je = j0 + v; int j2 = je + v; for (; j0 < je; ++j0, ++j2) { mint t0 = a[j0], t1 = a[j0 + v] * xx, t2 = a[j2] * ww, t3 = a[j2 + v] * wx; mint t0p2 = t0 + t2, t1p3 = t1 + t3; mint t0m2 = t0 - t2, t1m3 = (t1 - t3) * imag; a[j0] = t0p2 + t1p3, a[j0 + v] = t0p2 - t1p3; a[j2] = t0m2 + t1m3, a[j2 + v] = t0m2 - t1m3; } xx *= dw[__builtin_ctzll((jh += 4))]; } u <<= 2; v >>= 2; } } void ifft4(vector &a, int k) { if ((int)a.size() <= 1) return; if (k == 1) { mint a1 = a[1]; a[1] = a[0] - a[1]; a[0] = a[0] + a1; return; } int u = 1 << (k - 2); int v = 1; mint one = mint(1); mint imag = dy[1]; while (u) { // jh = 0 { int j0 = 0; int j1 = v; int j2 = v + v; int j3 = j2 + v; for (; j0 < v; ++j0, ++j1, ++j2, ++j3) { mint t0 = a[j0], t1 = a[j1], t2 = a[j2], t3 = a[j3]; mint t0p1 = t0 + t1, t2p3 = t2 + t3; mint t0m1 = t0 - t1, t2m3 = (t2 - t3) * imag; a[j0] = t0p1 + t2p3, a[j2] = t0p1 - t2p3; a[j1] = t0m1 + t2m3, a[j3] = t0m1 - t2m3; } } // jh >= 1 mint ww = one, xx = one * dy[2], yy = one; u <<= 2; for (int jh = 4; jh < u;) { ww = xx * xx, yy = xx * imag; int j0 = jh * v; int je = j0 + v; int j2 = je + v; for (; j0 < je; ++j0, ++j2) { mint t0 = a[j0], t1 = a[j0 + v], t2 = a[j2], t3 = a[j2 + v]; mint t0p1 = t0 + t1, t2p3 = t2 + t3; mint t0m1 = (t0 - t1) * xx, t2m3 = (t2 - t3) * yy; a[j0] = t0p1 + t2p3, a[j2] = (t0p1 - t2p3) * ww; a[j0 + v] = t0m1 + t2m3, a[j2 + v] = (t0m1 - t2m3) * ww; } xx *= dy[__builtin_ctzll(jh += 4)]; } u >>= 4; v <<= 2; } if (k & 1) { u = 1 << (k - 1); for (int j = 0; j < u; ++j) { mint ajv = a[j] - a[j + u]; a[j] += a[j + u]; a[j + u] = ajv; } } } void ntt(vector &a) { if ((int)a.size() <= 1) return; fft4(a, __builtin_ctz(a.size())); } void intt(vector &a) { if ((int)a.size() <= 1) return; ifft4(a, __builtin_ctz(a.size())); mint iv = mint(a.size()).inverse(); for (auto &x : a) x *= iv; } vector multiply(const vector &a, const vector &b) { int l = a.size() + b.size() - 1; if (min(a.size(), b.size()) <= 40) { vector s(l); for (int i = 0; i < (int)a.size(); ++i) for (int j = 0; j < (int)b.size(); ++j) s[i + j] += a[i] * b[j]; return s; } int k = 2, M = 4; while (M < l) M <<= 1, ++k; setwy(k); vector s(M); for (int i = 0; i < (int)a.size(); ++i) s[i] = a[i]; fft4(s, k); if (a.size() == b.size() && a == b) { for (int i = 0; i < M; ++i) s[i] *= s[i]; } else { vector t(M); for (int i = 0; i < (int)b.size(); ++i) t[i] = b[i]; fft4(t, k); for (int i = 0; i < M; ++i) s[i] *= t[i]; } ifft4(s, k); s.resize(l); mint invm = mint(M).inverse(); for (int i = 0; i < l; ++i) s[i] *= invm; return s; } void ntt_doubling(vector &a) { int M = (int)a.size(); auto b = a; intt(b); mint r = 1, zeta = mint(pr).pow((mint::get_mod() - 1) / (M << 1)); for (int i = 0; i < M; i++) b[i] *= r, r *= zeta; ntt(b); copy(begin(b), end(b), back_inserter(a)); } }; template void FormalPowerSeries::set_fft() { if (!ntt_ptr) ntt_ptr = new NTT; } template FormalPowerSeries& FormalPowerSeries::operator*=( const FormalPowerSeries& r) { if (this->empty() || r.empty()) { this->clear(); return *this; } set_fft(); auto ret = static_cast*>(ntt_ptr)->multiply(*this, r); return *this = FormalPowerSeries(ret.begin(), ret.end()); } template void FormalPowerSeries::ntt() { set_fft(); static_cast*>(ntt_ptr)->ntt(*this); } template void FormalPowerSeries::intt() { set_fft(); static_cast*>(ntt_ptr)->intt(*this); } template void FormalPowerSeries::ntt_doubling() { set_fft(); static_cast*>(ntt_ptr)->ntt_doubling(*this); } template int FormalPowerSeries::ntt_pr() { set_fft(); return static_cast*>(ntt_ptr)->pr; } template FormalPowerSeries FormalPowerSeries::inv(int deg) const { assert((*this)[0] != mint(0)); if (deg == -1) deg = (int)this->size(); FormalPowerSeries res(deg); res[0] = {mint(1) / (*this)[0]}; for (int d = 1; d < deg; d <<= 1) { FormalPowerSeries f(2 * d), g(2 * d); for (int j = 0; j < min((int)this->size(), 2 * d); j++) f[j] = (*this)[j]; for (int j = 0; j < d; j++) g[j] = res[j]; f.ntt(); g.ntt(); for (int j = 0; j < 2 * d; j++) f[j] *= g[j]; f.intt(); for (int j = 0; j < d; j++) f[j] = 0; f.ntt(); for (int j = 0; j < 2 * d; j++) f[j] *= g[j]; f.intt(); for (int j = d; j < min(2 * d, deg); j++) res[j] = -f[j]; } return res.pre(deg); } template FormalPowerSeries FormalPowerSeries::exp(int deg) const { using fps = FormalPowerSeries; assert((*this).size() == 0 || (*this)[0] == mint(0)); if (deg == -1) deg = this->size(); fps inv; inv.reserve(deg + 1); inv.push_back(mint(0)); inv.push_back(mint(1)); auto inplace_integral = [&](fps& F) -> void { const int n = (int)F.size(); auto mod = mint::get_mod(); while ((int)inv.size() <= n) { int i = inv.size(); inv.push_back((-inv[mod % i]) * (mod / i)); } F.insert(begin(F), mint(0)); for (int i = 1; i <= n; i++) F[i] *= inv[i]; }; auto inplace_diff = [](fps& F) -> void { if (F.empty()) return; F.erase(begin(F)); mint coeff = 1, one = 1; for (int i = 0; i < (int)F.size(); i++) { F[i] *= coeff; coeff += one; } }; fps b{1, 1 < (int)this->size() ? (*this)[1] : 0}, c{1}, z1, z2{1, 1}; for (int m = 2; m < deg; m *= 2) { auto y = b; y.resize(2 * m); y.ntt(); z1 = z2; fps z(m); for (int i = 0; i < m; ++i) z[i] = y[i] * z1[i]; z.intt(); fill(begin(z), begin(z) + m / 2, mint(0)); z.ntt(); for (int i = 0; i < m; ++i) z[i] *= -z1[i]; z.intt(); c.insert(end(c), begin(z) + m / 2, end(z)); z2 = c; z2.resize(2 * m); z2.ntt(); fps x(begin(*this), begin(*this) + min(this->size(), m)); x.resize(m); inplace_diff(x); x.push_back(mint(0)); x.ntt(); for (int i = 0; i < m; ++i) x[i] *= y[i]; x.intt(); x -= b.diff(); x.resize(2 * m); for (int i = 0; i < m - 1; ++i) x[m + i] = x[i], x[i] = mint(0); x.ntt(); for (int i = 0; i < 2 * m; ++i) x[i] *= z2[i]; x.intt(); x.pop_back(); inplace_integral(x); for (int i = m; i < min(this->size(), 2 * m); ++i) x[i] += (*this)[i]; fill(begin(x), begin(x) + m, mint(0)); x.ntt(); for (int i = 0; i < 2 * m; ++i) x[i] *= y[i]; x.intt(); b.insert(end(b), begin(x) + m, end(x)); } return fps{begin(b), begin(b) + deg}; } /** * @brief NTT mod用FPSライブラリ * @docs docs/fps/ntt-friendly-fps.md */ template struct LazyMontgomeryModInt { using mint = LazyMontgomeryModInt; using i32 = int32_t; using u32 = uint32_t; using u64 = uint64_t; static constexpr u32 get_r() { u32 ret = mod; for (i32 i = 0; i < 4; ++i) ret *= 2 - mod * ret; return ret; } static constexpr u32 r = get_r(); static constexpr u32 n2 = -u64(mod) % mod; static_assert(mod < (1 << 30), "invalid, mod >= 2 ^ 30"); static_assert((mod & 1) == 1, "invalid, mod % 2 == 0"); static_assert(r * mod == 1, "this code has bugs."); u32 a; constexpr LazyMontgomeryModInt() : a(0) {} constexpr LazyMontgomeryModInt(const int64_t &b) : a(reduce(u64(b % mod + mod) * n2)){}; static constexpr u32 reduce(const u64 &b) { return (b + u64(u32(b) * u32(-r)) * mod) >> 32; } constexpr mint &operator+=(const mint &b) { if (i32(a += b.a - 2 * mod) < 0) a += 2 * mod; return *this; } constexpr mint &operator-=(const mint &b) { if (i32(a -= b.a) < 0) a += 2 * mod; return *this; } constexpr mint &operator*=(const mint &b) { a = reduce(u64(a) * b.a); return *this; } constexpr mint &operator/=(const mint &b) { *this *= b.inverse(); return *this; } constexpr mint operator+(const mint &b) const { return mint(*this) += b; } constexpr mint operator-(const mint &b) const { return mint(*this) -= b; } constexpr mint operator*(const mint &b) const { return mint(*this) *= b; } constexpr mint operator/(const mint &b) const { return mint(*this) /= b; } constexpr bool operator==(const mint &b) const { return (a >= mod ? a - mod : a) == (b.a >= mod ? b.a - mod : b.a); } constexpr bool operator!=(const mint &b) const { return (a >= mod ? a - mod : a) != (b.a >= mod ? b.a - mod : b.a); } constexpr mint operator-() const { return mint() - mint(*this); } constexpr mint operator+() const { return mint(*this); } constexpr mint pow(u64 n) const { mint ret(1), mul(*this); while (n > 0) { if (n & 1) ret *= mul; mul *= mul; n >>= 1; } return ret; } constexpr mint inverse() const { int x = get(), y = mod, u = 1, v = 0, t = 0, tmp = 0; while (y > 0) { t = x / y; x -= t * y, u -= t * v; tmp = x, x = y, y = tmp; tmp = u, u = v, v = tmp; } return mint{u}; } friend ostream &operator<<(ostream &os, const mint &b) { return os << b.get(); } friend istream &operator>>(istream &is, mint &b) { int64_t t; is >> t; b = LazyMontgomeryModInt(t); return (is); } constexpr u32 get() const { u32 ret = reduce(a); return ret >= mod ? ret - mod : ret; } static constexpr u32 get_mod() { return mod; } }; // #include "fps/arbitrary-fps.hpp" // using namespace Nyaan; using mint = LazyMontgomeryModInt<998244353>; // using mint = LazyMontgomeryModInt<1000000007>; using vm = vector; using vvm = vector; Binomial C; using fps = FormalPowerSeries; using namespace Nyaan; mint naive(ll N, ll M) { mint ans = 0; rep1(x, M) { ans += C(2 * x - 2, N - 1); } return ans; } mint calc(ll N, ll M) { // C(0,N-1), C(2, N-1), ..., C(2M-2, N-1) int lower = (N - 1 + 1) / 2; if(lower>=M)return 0; /* mint ans = 0; reg(i, lower, M) ans += C(i * 2, N - 1); return ans; */ vm a; mint ans = 0; reg(i, lower, lower + 13) ans += C(i * 2, N - 1), a.push_back(ans); return kth_term_of_p_recursive(a, M - lower - 1); } void q() { /** rep(i, 100) rep(j, 100) { trc(i, j, naive(i, j), calc(i, j)); assert(naive(i, j) == calc(i, j)); } //*/ inl(T); while (T--) { inl(N, M); out(calc(N, M)); } } void Nyaan::solve() { int t = 1; // in(t); while (t--) q(); }