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\n#include \n#include \n#include \n#include \n#include \n#include \n#include \n#include \n#include \n\nnamespace lib {\n\ntemplate \nclass Mint {\n static_assert(static_cast(Mod) > 1);\n unsigned v_;\n\n public:\n constexpr Mint() : v_(0) {}\n constexpr Mint(int v) : v_(safe_mod(v)) {}\n ~Mint() = default;\n static constexpr unsigned safe_mod(int v) {\n v %= static_cast(Mod);\n return static_cast(v < 0 ? v + static_cast(Mod) : v);\n }\n static constexpr unsigned mod() { return Mod; }\n constexpr unsigned val() const { return v_; }\n constexpr Mint &operator=(const Mint &) = default;\n constexpr Mint operator-() const { return Mint(Mod - v_); }\n constexpr Mint pow(int e) const {\n if (e < 0) return inv().pow(-e);\n for (Mint x(*this), res(1);; x *= x) {\n if (e & 1) res *= x;\n if ((e >>= 1) == 0) return res;\n }\n }\n constexpr Mint inv() const {\n int x1 = 1, x3 = 0, a = static_cast(v_), b = static_cast(Mod);\n while (b != 0) {\n int q = a / b, x1_old = x1, a_old = a;\n x1 = x3, x3 = x1_old - x3 * q, a = b, b = a_old - b * q;\n }\n return Mint(x1);\n }\n constexpr Mint &operator+=(const Mint &rhs) {\n if ((v_ += rhs.v_) >= Mod) v_ -= Mod;\n return *this;\n }\n constexpr Mint &operator-=(const Mint &rhs) {\n if ((v_ += Mod - rhs.v_) >= Mod) v_ -= Mod;\n return *this;\n }\n constexpr Mint &operator*=(const Mint &rhs) {\n v_ = static_cast(v_) * rhs.v_ % Mod;\n return *this;\n }\n constexpr Mint &operator/=(const Mint &rhs) { return operator*=(rhs.inv()); }\n void swap(Mint &rhs) {\n unsigned int v = v_;\n v_ = rhs.v_, rhs.v_ = v;\n }\n\n friend constexpr Mint operator+(const Mint &lhs, const Mint &rhs) {\n return Mint(lhs) += rhs;\n }\n friend constexpr Mint operator-(const Mint &lhs, const Mint &rhs) {\n return Mint(lhs) -= rhs;\n }\n friend constexpr Mint operator*(const Mint &lhs, const Mint &rhs) {\n return Mint(lhs) *= rhs;\n }\n friend constexpr Mint operator/(const Mint &lhs, const Mint &rhs) {\n return Mint(lhs) /= rhs;\n }\n friend constexpr bool operator==(const Mint &lhs, const Mint &rhs) {\n return lhs.v_ == rhs.v_;\n }\n friend constexpr bool operator!=(const Mint &lhs, const Mint &rhs) {\n return lhs.v_ != rhs.v_;\n }\n friend std::istream &operator>>(std::istream &lhs, Mint &rhs) {\n int v;\n lhs >> v;\n rhs = Mint(v);\n return lhs;\n }\n friend std::ostream &operator<<(std::ostream &lhs, const Mint &rhs) {\n return lhs << rhs.v_;\n }\n};\n\nnamespace detail {\n\nconstexpr unsigned pow_mod(unsigned a, unsigned e, const unsigned p) {\n for (unsigned res = 1;; a = static_cast(a) * a % p) {\n if (e & 1) res = static_cast(res) * a % p;\n if ((e >>= 1) == 0) return res;\n }\n}\n\nconstexpr unsigned quadratic_nonresidue(const unsigned p) {\n // `p` must be an odd prime\n for (unsigned i = 2;; ++i)\n if (pow_mod(i, p >> 1, p) == p - 1) return i;\n}\n\ntemplate \nconstexpr int bsf(T v) {\n if (static_cast>(v) <= 0) return -1;\n int res = 0;\n for (; (v & 1) == 0; ++res) v >>= 1;\n return res;\n}\n\nconstexpr unsigned gen_of_2_Sylow_subgroup(const unsigned p) {\n return pow_mod(quadratic_nonresidue(p), p >> bsf(p - 1), p);\n}\n\ntemplate \nconstexpr std::array root() {\n std::array rt; // order(`rt[i]`) = 2^(i + 2)\n rt.back() = T(gen_of_2_Sylow_subgroup(T::mod()));\n for (int i = bsf(T::mod() - 1) - 3; i >= 0; --i)\n rt[i] = rt[i + 1] * rt[i + 1];\n return rt;\n}\n\ntemplate \nconstexpr std::array iroot() {\n std::array irt;\n irt.back() = T(gen_of_2_Sylow_subgroup(T::mod())).inv();\n for (int i = bsf(T::mod() - 1) - 3; i >= 0; --i)\n irt[i] = irt[i + 1] * irt[i + 1];\n return irt;\n}\n\ntemplate \nstd::vector sqrt_mod(T a) { // don't use\n const T ONE(1), MINUS_ONE(-1), ZERO(0);\n if (a == ZERO) return {ZERO};\n if (a.pow(T::mod() >> 1) == MINUS_ONE) return {};\n if ((T::mod() & 3) == 3) {\n T res(a.pow((T::mod() + 1) >> 2));\n return {res, -res};\n }\n\n std::mt19937 gen(std::random_device{}());\n std::uniform_int_distribution dis(2, T::mod() - 1);\n\n const T f(a * T(4));\n\n T t, w2;\n do {\n t = T(dis(gen)), w2 = t * t - f;\n if (w2 == ZERO) {\n t /= T(2);\n return {t, -t};\n }\n } while (w2.pow(T::mod() >> 1) != MINUS_ONE);\n\n T k0(ONE), k1, k2(-t), k3(a);\n for (auto e = (T::mod() + 1) >> 1;;) {\n if (e & 1) {\n k0 = k1 - k0 * k2, k1 *= k3;\n } else {\n k1 = k0 * k3 - k1 * k2;\n }\n k2 = k3 + k3 - k2 * k2, k3 *= k3;\n if ((e >>= 1) == 0) return {k0, -k0};\n }\n}\n\n} // namespace detail\n\nint ntt_len(int n) {\n --n;\n n |= n >> 1;\n n |= n >> 2;\n n |= n >> 4;\n n |= n >> 8;\n return (n | n >> 16) + 1;\n}\n\ntemplate \nvoid dft_n(Iter a, int n) {\n assert((n & (n - 1)) == 0);\n using T = typename std::iterator_traits::value_type;\n static constexpr auto rt = detail::root();\n static std::vector root(1);\n if (int s = static_cast(root.size()); s << 1 < n) {\n root.resize(n >> 1);\n for (int i = detail::bsf(s); (1 << i) < (n >> 1); ++i) {\n int j = 1 << i;\n root[j] = rt[i];\n for (int k = j + 1; k < (j << 1); ++k) root[k] = root[k - j] * root[j];\n }\n }\n for (int j = 0, l = n >> 1; j != l; ++j) {\n T u(a[j]), v(a[j + l]);\n a[j] = u + v, a[j + l] = u - v;\n }\n for (int i = n >> 1; i >= 2; i >>= 1) {\n for (int j = 0, l = i >> 1; j != l; ++j) {\n T u(a[j]), v(a[j + l]);\n a[j] = u + v, a[j + l] = u - v;\n }\n for (int j = i, l = i >> 1, m = 1; j != n; j += i, ++m) {\n for (int k = j; k != j + l; ++k) {\n T u(a[k]), v(a[k + l] * root[m]);\n a[k] = u + v, a[k + l] = u - v;\n }\n }\n }\n}\n\ntemplate \nvoid idft_n(Iter a, int n) {\n assert((n & (n - 1)) == 0);\n using T = typename std::iterator_traits::value_type;\n static constexpr auto rt = detail::iroot();\n static std::vector root(1);\n if (int s = static_cast(root.size()); s << 1 < n) {\n root.resize(n >> 1);\n for (int i = detail::bsf(s); (1 << i) < (n >> 1); ++i) {\n int j = 1 << i;\n root[j] = rt[i];\n for (int k = j + 1; k < (j << 1); ++k) root[k] = root[k - j] * root[j];\n }\n }\n for (int i = 2; i < n; i <<= 1) {\n for (int j = 0, l = i >> 1; j != l; ++j) {\n T u(a[j]), v(a[j + l]);\n a[j] = u + v, a[j + l] = u - v;\n }\n for (int j = i, l = i >> 1, m = 1; j != n; j += i, ++m) {\n for (int k = j; k != j + l; ++k) {\n T u(a[k]), v(a[k + l]);\n a[k] = u + v, a[k + l] = (u - v) * root[m];\n }\n }\n }\n const T iv(T::mod() - T::mod() / n);\n for (int j = 0, l = n >> 1; j != l; ++j) {\n T u(a[j] * iv), v(a[j + l] * iv);\n a[j] = u + v, a[j + l] = u - v;\n }\n}\n\ntemplate \nclass RConv { // O(n log^2 n) impl\n std::vector a_, b_, c_; // `a * b = c`\n std::vector> ac_, bc_; // cached DFTs\n std::function ha_, hb_; // handle for `a` and `b`\n int n_; // counter\n\n template \n static auto wrap(T0 &&f, int &n, const std::vector &c, std::vector &e) {\n if constexpr (std::is_invocable_r_v &>) {\n return std::bind(\n [f](int n, const std::vector &c, std::vector &e) mutable {\n return T(e.emplace_back(f(n, c)));\n },\n std::cref(n), std::cref(c), std::ref(e));\n } else if constexpr (std::is_invocable_r_v) {\n return std::bind(\n [f](int n, std::vector &e) mutable {\n return T(e.emplace_back(f(n)));\n },\n std::cref(n), std::ref(e));\n } else if constexpr (std::is_invocable_r_v) {\n return std::bind(\n [f](std::vector &e) mutable { return T(e.emplace_back(f())); },\n std::ref(e));\n } else {\n throw;\n }\n }\n\n public:\n // `h0` multiplicand, `h1` multiplier\n template \n RConv(F1 &&h0, F2 &&h1)\n : a_(),\n b_(),\n c_(4, T(0)),\n ac_(),\n bc_(),\n ha_(wrap(h0, n_, c_, a_)),\n hb_(wrap(h1, n_, c_, b_)),\n n_(0) {}\n ~RConv() = default;\n\n const std::vector &multiplicand() const { return a_; }\n const std::vector &multiplier() const { return b_; }\n T at(int k) { return operator[](k); }\n T operator[](int k) {\n while (n_ <= k) next();\n return c_[k];\n }\n\n T next() {\n static constexpr int BASE_CASE_SIZE = 32;\n int l = ntt_len(n_ << 1 | 1);\n if (static_cast(c_.size()) < l) c_.resize(l);\n switch (n_) {\n case 0:\n c_[0] = ha_() * hb_();\n break;\n case 1:\n c_[1] = ha_() * b_.front() + a_.front() * hb_();\n c_[2] = a_[1] * b_[1];\n break;\n case 2:\n c_[2] += ha_() * b_.front() + a_.front() * hb_();\n c_[3] = a_[2] * b_[1] + a_[1] * b_[2];\n break;\n default:\n if ((n_ & (n_ - 1)) == 0) {\n int t0 = n_ >> 1, t1 = n_;\n auto &c0 = ac_.emplace_back(a_.begin() + t0, a_.begin() + t1);\n auto &c1 = bc_.emplace_back(b_.begin() + t0, b_.begin() + t1);\n c0.resize(t1, T(0));\n c1.resize(t1, T(0));\n dft_n(c0.begin(), t1), dft_n(c1.begin(), t1);\n std::vector res(c0);\n for (int i = 0; i < t1; ++i) res[i] *= c1[i];\n idft_n(res.begin(), t1);\n for (int i = 0; i < t1 - 1; ++i) c_[t1 + i] += res[i];\n }\n c_[n_] += ha_() * b_.front() + a_.front() * hb_();\n c_[n_ + 1] += a_[1] * b_.back() + a_.back() * b_[1];\n for (int sft = 1, offset = ntt_len(n_ + 1) >> 1, t = n_ + 1 - offset;\n (t & 1) == 0 && 1 << sft < offset; ++sft, t >>= 1)\n if (1 << sft <= BASE_CASE_SIZE) {\n for (int i = 0, m = n_ + 1 - (1 << sft); i != 1 << sft; ++i)\n for (int j = 0; j != 1 << sft; ++j)\n c_[n_ + 1 + i + j] += a_[m + i] * b_[j + (1 << sft)] +\n a_[j + (1 << sft)] * b_[m + i];\n } else {\n std::vector c0(a_.begin() + n_ + 1 - (1 << sft),\n a_.begin() + n_ + 1);\n std::vector c1(b_.begin() + n_ + 1 - (1 << sft),\n b_.begin() + n_ + 1);\n c0.resize(2 << sft, T(0));\n c1.resize(2 << sft, T(0));\n dft_n(c0.begin(), 2 << sft), dft_n(c1.begin(), 2 << sft);\n for (int i = 0; i != 2 << sft; ++i)\n c0[i] = c0[i] * bc_[sft - 1][i] + c1[i] * ac_[sft - 1][i];\n idft_n(c0.begin(), 2 << sft);\n for (int i = 0; i != (2 << sft) - 1; ++i) c_[n_ + 1 + i] += c0[i];\n }\n }\n return c_[n_++];\n }\n};\n\ntemplate \nclass Invs {\n std::vector ivs;\n\n public:\n Invs() : ivs(1) {}\n ~Invs() = default;\n T operator()(int k) { // preprocess modular inverse from 1 to k\n if (int n = static_cast(ivs.size()); n <= k) {\n int nn = n;\n while (nn <= k) nn <<= 1;\n ivs.resize(nn);\n T v(1);\n for (int i = n; i != nn; ++i) ivs[i] = v, v *= T(i);\n v = v.inv();\n for (int i = nn - 1; i >= n; --i) ivs[i] *= v, v *= T(i);\n }\n return ivs[k];\n }\n};\n\ntemplate \nclass FPS {\n using F = std::function;\n F h_;\n\n static inline Invs invs;\n\n public:\n class Resetter {\n std::shared_ptr p_;\n\n public:\n explicit Resetter(std::shared_ptr p) : p_(p) {}\n ~Resetter() = default;\n Resetter &operator=(const Resetter &) = default;\n\n void set(const FPS &rhs) { *p_ = rhs.handle(); }\n };\n\n class Idm { // indeterminate\n int p_;\n\n public:\n explicit Idm(int p) : p_(p) {}\n explicit Idm() : Idm(1) {}\n Idm operator^(int p) const { return Idm(p_ * p); }\n Idm operator*(const Idm &rhs) const { return Idm(p_ + rhs.p_); }\n int pow() const { return p_; }\n };\n\n FPS() : h_([](int) { return T(0); }) {}\n FPS(F f)\n : h_([f, cache = std::make_shared>()](int k) -> T {\n for (int i = static_cast(cache->size()); i <= k; ++i)\n cache->emplace_back(f(i));\n return cache->at(k);\n }) {}\n FPS(const std::vector &coeff)\n : h_([cache = std::make_shared>(coeff)](int k) -> T {\n return k < static_cast(cache->size()) ? cache->at(k) : T(0);\n }) {}\n FPS(T v) : h_([v](int k) { return k == 0 ? v : T(0); }) {}\n ~FPS() = default;\n FPS(const FPS &) = default;\n FPS &operator=(const FPS &) = default;\n F handle() const { return h_; }\n static Idm idm() { return Idm(); }\n Resetter reset() {\n auto p = std::make_shared();\n h_ = [p](int i) { return (*p)(i); };\n return Resetter(p);\n }\n\n T operator[](int k) const { return h_(k); }\n /*------------------- helper functions begin -------------------*/\n FPS operator()(const Idm &rhs) const { return scale(rhs.pow()); }\n FPS operator*(const Idm &rhs) const { return shift(rhs.pow()); }\n FPS operator/(const Idm &rhs) const { return shift(-rhs.pow()); }\n FPS operator*(const T &rhs) const {\n return FPS([h = h_, v = rhs](int i) { return h(i) * v; });\n }\n FPS operator/(const T &rhs) const {\n // `rhs == 0` is not allowed\n return FPS([h = h_, v = rhs.inv()](int i) { return h(i) * v; });\n }\n FPS operator+(const T &rhs) const {\n return FPS([h = h_, v = rhs](int i) { return i == 0 ? h(i) + v : h(i); });\n }\n FPS operator-(const T &rhs) const {\n return FPS([h = h_, v = rhs](int i) { return i == 0 ? h(i) - v : h(i); });\n }\n friend FPS operator*(const Idm &lhs, const FPS &rhs) {\n return rhs.shift(lhs.pow());\n }\n friend FPS operator/(const Idm &lhs, const FPS &rhs) {\n return lhs * rhs.inv();\n }\n friend FPS operator-(const T &lhs, const FPS &rhs) {\n return FPS(\n [h = rhs.h_, v = lhs](int i) { return i == 0 ? v - h(i) : -h(i); });\n }\n friend FPS operator+(const T &lhs, const FPS &rhs) {\n return FPS(\n [h = rhs.h_, v = lhs](int i) { return i == 0 ? h(i) + v : h(i); });\n }\n friend FPS operator*(const T &lhs, const FPS &rhs) {\n return FPS([h = rhs.h_, v = lhs](int i) { return h(i) * v; });\n }\n friend FPS operator/(const T &lhs, const FPS &rhs) { return lhs * rhs.inv(); }\n /*------------------- helper functions end -------------------*/\n FPS scale(int k) const {\n return FPS([h = h_, k](int i) { return i % k == 0 ? h(i / k) : T(0); });\n }\n FPS shift(int k) const {\n return FPS([h = h_, k](int i) { return i - k < 0 ? T(0) : h(i - k); });\n }\n FPS deriv() const {\n return FPS([h = h_](int i) { return h(i + 1) * T(i + 1); });\n }\n FPS integr(T c = T(0)) const {\n return FPS([h = h_, c](int i) { return i == 0 ? c : h(i - 1) * invs(i); });\n }\n FPS inv() const {\n auto rc = std::make_shared>(\n [h = h_](int i) { return h(i); },\n [h = h_, iv = T()](int i, const auto &c) mutable {\n return i == 0 ? T(iv = h(0).inv()) : -(c[i] + h(i) * iv) * iv;\n });\n return FPS(\n [rc](int i) { return rc->next(), rc->multiplier()[i]; }); // safe\n }\n FPS exp() const {\n auto rc = std::make_shared>(\n [h = h_](int i) { return h(i + 1) * T(i + 1); },\n [](int i, const auto &c) {\n return i == 0 ? T(1) : c[i - 1] * invs(i);\n });\n return FPS([rc](int i) { return i == 0 ? T(1) : rc->at(i - 1) * invs(i); });\n }\n FPS log() const { return (deriv() / (*this)).integr(); }\n FPS pow(int k) const {\n return FPS([h = h_, kk = T(k), k, zero_cnt = 0ull,\n s = std::optional()](int i) mutable {\n if (s)\n return static_cast(i) < zero_cnt\n ? T(0)\n : (*s)(i - static_cast(zero_cnt));\n T v(h(i));\n if (v == T(0)) return ++zero_cnt, T(0);\n zero_cnt *= k;\n FPS t0([os = i, iv = v.inv(), h](int i) { return h(i + os) * iv; });\n FPS t1([h0 = t0.log().handle(), kk](int i) { return h0(i) * kk; });\n s.emplace([vk = v.pow(k), h1 = t1.exp().handle()](int i) {\n return h1(i) * vk;\n });\n return zero_cnt == 0 ? (*s)(i) : T(0);\n });\n }\n FPS sqrt(std::function f) const {\n // `h_(0) == 0` is not allowed\n auto t = [h = h_, f, i2 = T()](int i, auto const &c) mutable {\n if (i != 0) return (h(i) - c[i]) * i2;\n T fi(f(h(i)));\n i2 = (fi * T(2)).inv();\n return fi;\n };\n auto rc = std::make_shared>(t, t);\n return FPS([rc](int i) { return rc->next(), rc->multiplier()[i]; });\n }\n FPS operator+(const FPS &rhs) const {\n return FPS([h0 = h_, h1 = rhs.h_](int i) { return h0(i) + h1(i); });\n }\n FPS operator-(const FPS &rhs) const {\n return FPS([h0 = h_, h1 = rhs.h_](int i) { return h0(i) - h1(i); });\n }\n FPS operator-() const {\n return FPS([h = h_](int i) { return -h(i); });\n }\n FPS operator*(const FPS &rhs) const {\n auto rc = std::make_shared>([h = h_](int i) { return h(i); },\n [h = rhs.h_](int i) { return h(i); });\n return FPS([rc](int) { return rc->next(); });\n }\n FPS operator/(const FPS &rhs) const {\n auto rc = std::make_shared>(\n [h0 = h_, h1 = rhs.h_, iv = T(), t0 = T()](int i,\n const auto &c) mutable {\n if (i == 0) t0 = h0(0) * (iv = h1(0).inv());\n return i == 0 ? t0 : (h0(i) - h1(i) * t0 - c[i]) * iv;\n },\n [h = rhs.h_](int i) { return h(i); });\n return FPS([rc](int i) { return rc->next(), rc->multiplicand()[i]; });\n }\n /*------------------- Pólya operators begin -------------------*/\n FPS Q() const { // SEQUENCE(SEQ)\n // `h_(0) == 0` is not allowed\n return FPS([h = h_](int i) { return i == 0 ? T(1) : -h(i); }).inv();\n }\n FPS Exp() const { // MULTISET(MSET)\n // `h_(0) == 0` is not allowed\n FPS res([h = h_, cache = std::make_shared>()](int i) {\n if (i == 0) return T(0);\n if ((i & (i - 1)) == 0) {\n cache->resize(i << 1, T(0));\n for (int j = 1; j < i; ++j) {\n T hj(h(j));\n for (int k = 2; j * k < i << 1; ++k)\n if (j * k >= i) cache->at(j * k) += hj * invs(k);\n }\n }\n return T(cache->at(i) += h(i));\n });\n return res.exp();\n }\n FPS Exp_m() const { // POWERSET(PSET)\n // `h_(0) == 0` is not allowed\n FPS res([h = h_, cache = std::make_shared>()](int i) {\n if (i == 0) return T(0);\n if ((i & (i - 1)) == 0) {\n cache->resize(i << 1, T(0));\n for (int j = 1; j < i; ++j) {\n T hj(h(j));\n for (int k = 2; j * k < i << 1; ++k)\n if (j * k >= i) {\n if (k & 1) {\n cache->at(j * k) += hj * invs(k);\n } else {\n cache->at(j * k) -= hj * invs(k);\n }\n }\n }\n }\n return T(cache->at(i) += h(i));\n });\n return res.exp();\n }\n /*------------------- Pólya operators end ---------------------*/\n};\n\n} // namespace lib\n\n#include \nint main() {\n using lib::Mint;\n using lib::FPS;\n int n, k;\n std::cin >> n >> k;\n FPS> A([](int) {\n int n;\n std::cin >> n;\n return n;\n }), B(A.pow(k));\n for (int i = 0; i != n; ++i) std::cout << B[i] << ' ';\n}","codes",[],"stdin","2 2\n1 1","compiler","warning,boost-nothing-gcc-head,c++17,cpp-no-pedantic","createdAt",1645370718,"title","description","isPrivate","compilerInfo",{"_11":332,"_13":3399,"_15":333,"_17":18,"_19":3400,"_22":23,"_24":25,"_26":335,"_28":3401},"12.0.1 20220219 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