#include #define REP_(i, a_, b_, a, b, ...) \ for (int i = (a), END_##i = (b); i < END_##i; ++i) #define REP(i, ...) REP_(i, __VA_ARGS__, __VA_ARGS__, 0, __VA_ARGS__) #define ALL(x) std::begin(x), std::end(x) using i64 = long long; template inline bool chmax(T &a, U b) { return a < b and ((a = std::move(b)), true); } template inline bool chmin(T &a, U b) { return a > b and ((a = std::move(b)), true); } template inline int ssize(const T &a) { return (int) a.size(); } template inline std::ostream &print_one(const T &x, char endc) { if constexpr (std::is_same_v) { return std::cout << (x ? "Yes" : "No") << endc; } else { return std::cout << x << endc; } } template inline std::ostream &print(const T &x) { return print_one(x, '\n'); } template std::ostream &print(const T &head, Ts... tail) { return print_one(head, ' '), print(tail...); } inline std::ostream &print() { return std::cout << '\n'; } template std::ostream &print_seq(const Container &a, std::string_view sep = " ", std::string_view ends = "\n", std::ostream &os = std::cout) { auto b = std::begin(a), e = std::end(a); for (auto it = std::begin(a); it != e; ++it) { if (it != b) os << sep; os << *it; } return os << ends; } template struct is_iterable : std::false_type {}; template struct is_iterable())), decltype(std::end(std::declval()))>> : std::true_type { }; template::value && !std::is_same::value>> std::ostream &operator<<(std::ostream &os, const T &a) { return print_seq(a, ", ", "", (os << "{")) << "}"; } struct CastInput { template operator T() const { T x; std::cin >> x; return x; } struct Sized { std::size_t n; template operator T() const { T x(n); for (auto &e: x) std::cin >> e; return x; } }; Sized operator()(std::size_t n) const { return {n}; } } const in; inline void check(bool cond, const char *message = "!ERROR!") { if (not cond) throw std::runtime_error(message); } #ifdef MY_DEBUG #include "debug_dump.hpp" #else #define DUMP(...) #define cerr if(false)std::cerr #endif using namespace std; template struct NaiveMult { using value_type = T; static constexpr int dmax() { return DMAX; } static std::vector multiply(const std::vector &x, const std::vector &y) { const int n = std::min(x.size() + y.size() - 1, DMAX + 1); const int mi = std::min(x.size(), n); std::vector res(n); for (int i = 0; i < mi; ++i) { for (int j = 0; j < int(y.size()); ++j) { if (i + j >= n) break; res[i + j] += x[i] * y[j]; } } return res; } static std::vector invert(const std::vector &x) { std::vector res(DMAX + 1); res[0] = x[0].inv(); for (int i = 1; i <= DMAX; ++i) { T s = 0; const int mj = std::min(i + 1, x.size()); for (int j = 1; j < mj; ++j) { s += x[j] * res[i - j]; } res[i] = -res[0] * s; } return res; } }; // Formal Power Series (dense format). template struct DenseFPS { using T = typename Mult::value_type; static constexpr int dmax() { return Mult::dmax(); } // Coefficients of terms from x^0 to x^DMAX. std::vector coeff_; DenseFPS() : coeff_(1, 0) {} // = 0 * x^0 explicit DenseFPS(std::vector c) : coeff_(std::move(c)) { while (size() > dmax() + 1) coeff_.pop_back(); assert(size() > 0); } DenseFPS(std::initializer_list init) : coeff_(init.begin(), init.end()) { while (size() > dmax() + 1) coeff_.pop_back(); assert(size() > 0); } DenseFPS(const DenseFPS &other) : coeff_(other.coeff_) {} DenseFPS(DenseFPS &&other) : coeff_(std::move(other.coeff_)) {} DenseFPS &operator=(const DenseFPS &other) { coeff_ = other.coeff_; return *this; } DenseFPS &operator=(DenseFPS &&other) { coeff_ = std::move(other.coeff_); return *this; } // size <= dmax + 1 inline int size() const { return static_cast(coeff_.size()); } // Returns the coefficient of x^k. inline T operator[](int k) const { return (k >= size()) ? 0 : coeff_[k]; } DenseFPS &operator+=(const T &scalar) { coeff_[0] += scalar; return *this; } friend DenseFPS operator+(const DenseFPS &f, const T &scalar) { return DenseFPS(f) += scalar; } DenseFPS &operator+=(const DenseFPS &other) { if (size() < other.size()) coeff_.resize(other.size()); for (int i = 0; i < other.size(); ++i) coeff_[i] += other[i]; return *this; } friend DenseFPS operator+(const DenseFPS &f, const DenseFPS &g) { return DenseFPS(f) += g; } DenseFPS &operator-=(const DenseFPS &other) { if (size() < other.size()) coeff_.resize(other.size()); for (int i = 0; i < other.size(); ++i) coeff_[i] -= other[i]; return *this; } friend DenseFPS operator-(const DenseFPS &f, const DenseFPS &g) { return DenseFPS(f) -= g; } DenseFPS operator-() const { return *this * -1; } DenseFPS &operator*=(const T &scalar) { for (auto &x: coeff_) x *= scalar; return *this; } friend DenseFPS operator*(const DenseFPS &f, const T &scalar) { return DenseFPS(f) *= scalar; } friend DenseFPS operator*(const T &scalar, const DenseFPS &g) { return DenseFPS{scalar} *= g; } DenseFPS &operator*=(const DenseFPS &other) { return *this = DenseFPS(Mult::multiply(std::move(this->coeff_), other.coeff_)); } friend DenseFPS operator*(const DenseFPS &f, const DenseFPS &g) { return DenseFPS(Mult::multiply(f.coeff_, g.coeff_)); } DenseFPS &operator/=(const T &scalar) { for (auto &x: coeff_) x /= scalar; return *this; } friend DenseFPS operator/(const DenseFPS &f, const T &scalar) { return DenseFPS(f) /= scalar; } friend DenseFPS operator/(const T &scalar, const DenseFPS &g) { return DenseFPS{scalar} /= g; } DenseFPS &operator/=(const DenseFPS &other) { return *this *= DenseFPS(Mult::invert(other.coeff_)); } friend DenseFPS operator/(const DenseFPS &f, const DenseFPS &g) { return f * DenseFPS(Mult::invert(g.coeff_)); } DenseFPS pow(i64 t) const { assert(t >= 0); DenseFPS res = {1}, base = *this; while (t) { if (t & 1) res *= base; base *= base; t >>= 1; } return res; } // Multiplies by (1 + c * x^k). void multiply2_inplace(int k, int c) { assert(k > 0); if (size() <= dmax()) { coeff_.resize(min(size() + k, dmax() + 1), 0); } for (int i = size() - 1; i >= k; --i) { coeff_[i] += coeff_[i - k] * c; } } // Multiplies by (1 + c * x^k). DenseFPS multiply2(int k, int c) const { DenseFPS res = *this; res.multiply2_inplace(k, c); return res; } // Divides by (1 + c * x^k). void divide2_inplace(int k, int c) { assert(k > 0); for (int i = k; i < size(); ++i) { coeff_[i] -= coeff_[i - k] * c; } } // Divides by (1 + c * x^k). DenseFPS divide2(int k, int c) const { DenseFPS res = *this; res.divide2_inplace(k, c); return res; } // Multiplies by x^k. void shift_inplace(int k) { if (k > 0) { if (size() <= dmax()) { coeff_.resize(min(size() + k, dmax() + 1), 0); } for (int i = size() - 1; i >= k; --i) { coeff_[i] = coeff_[i - k]; } for (int i = k - 1; i >= 0; --i) { coeff_[i] = 0; } } else if (k < 0) { k *= -1; for (int i = k; i < size(); ++i) { coeff_[i - k] = coeff_[i]; } for (int i = size() - k; i < size(); ++i) { // If coefficients of degrees higher than dmax() were truncated // beforehand, you lose the information. Ensure dmax() is big enough. coeff_[i] = 0; } } } // Multiplies by x^k. DenseFPS shift(int k) const { DenseFPS res = *this; res.shift_inplace(k); return res; } T eval(const T &a) const { T res = 0, x = 1; for (auto c: coeff_) { res += c * x; x *= a; } return res; } }; using Real = long double; constexpr int D = 60; using DF = DenseFPS>; auto solve() { int n = in, K = in; DF f = {1}, g = {1}; const Real d6 = 1 / 6.0; const Real d3 = 1 / 3.0; DF a = {0, d6, d6, d6, d6, d6, d6}; DF b = {0, 0, 0, 0, d3, d3, d3}; REP(i, n) { g *= a; } REP(i, K) { f *= b; } REP(i, n - K) { f *= a; } g.divide2_inplace(1, -1); // cumsum Real ans = 0; for (int i = 1; i <= D; ++i) { ans += f[i] * g[i - 1]; } return ans; } int main() { ios_base::sync_with_stdio(false), cin.tie(nullptr); cout << std::fixed << std::setprecision(18); const int T = 1;//in; REP(t, T) { auto ans = solve(); print(ans); } }