#include using namespace std; #define FOR(i, begin, end) for(int i=(begin),i##_end_=(end);i ostream &operator<<(ostream &os, const vector &vec){ os << "["; for (auto v : vec) os << v << ","; os << "]"; return os; } #define dbg(x) cerr << #x << " = " << (x) << " (L" << __LINE__ << ") " << __FILE__ << endl; constexpr int MOD = 1012924417; template struct ModInt { using lint = long long; static int get_mod() { return mod; } static int get_primitive_root() { static int primitive_root = 0; if (!primitive_root) { primitive_root = [&](){ std::set fac; int v = mod - 1; for (lint i = 2; i * i <= v; i++) while (v % i == 0) fac.insert(i), v /= i; if (v > 1) fac.insert(v); for (int g = 1; g < mod; g++) { bool ok = true; for (auto i : fac) if (ModInt(g).power((mod - 1) / i) == 1) { ok = false; break; } if (ok) return g; } return -1; }(); } return primitive_root; } int val; constexpr ModInt() : val(0) {} constexpr ModInt &_setval(lint v) { val = (v >= mod ? v - mod : v); return *this; } constexpr ModInt(lint v) { _setval(v % mod + mod); } explicit operator bool() const { return val != 0; } constexpr ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); } constexpr ModInt operator-(const ModInt &x) const { return ModInt()._setval((lint)val - x.val + mod); } constexpr ModInt operator*(const ModInt &x) const { return ModInt()._setval((lint)val * x.val % mod); } constexpr ModInt operator/(const ModInt &x) const { return ModInt()._setval((lint)val * x.inv() % mod); } constexpr ModInt operator-() const { return ModInt()._setval(mod - val); } constexpr ModInt &operator+=(const ModInt &x) { return *this = *this + x; } constexpr ModInt &operator-=(const ModInt &x) { return *this = *this - x; } constexpr ModInt &operator*=(const ModInt &x) { return *this = *this * x; } constexpr ModInt &operator/=(const ModInt &x) { return *this = *this / x; } friend constexpr ModInt operator+(lint a, const ModInt &x) { return ModInt()._setval(a % mod + x.val); } friend constexpr ModInt operator-(lint a, const ModInt &x) { return ModInt()._setval(a % mod - x.val + mod); } friend constexpr ModInt operator*(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.val % mod); } friend constexpr ModInt operator/(lint a, const ModInt &x) { return ModInt()._setval(a % mod * x.inv() % mod); } constexpr bool operator==(const ModInt &x) const { return val == x.val; } constexpr bool operator!=(const ModInt &x) const { return val != x.val; } bool operator<(const ModInt &x) const { return val < x.val; } // To use std::map friend std::istream &operator>>(std::istream &is, ModInt &x) { lint t; is >> t; x = ModInt(t); return is; } friend std::ostream &operator<<(std::ostream &os, const ModInt &x) { os << x.val; return os; } constexpr lint power(lint n) const { lint ans = 1, tmp = this->val; while (n) { if (n & 1) ans = ans * tmp % mod; tmp = tmp * tmp % mod; n /= 2; } return ans; } constexpr lint inv() const { return this->power(mod - 2); } constexpr ModInt operator^(lint n) const { return ModInt(this->power(n)); } constexpr ModInt &operator^=(lint n) { return *this = *this ^ n; } inline ModInt fac() const { static std::vector facs; int l0 = facs.size(); if (l0 > this->val) return facs[this->val]; facs.resize(this->val + 1); for (int i = l0; i <= this->val; i++) facs[i] = (i == 0 ? ModInt(1) : facs[i - 1] * ModInt(i)); return facs[this->val]; } ModInt doublefac() const { lint k = (this->val + 1) / 2; if (this->val & 1) return ModInt(k * 2).fac() / ModInt(2).power(k) / ModInt(k).fac(); else return ModInt(k).fac() * ModInt(2).power(k); } ModInt nCr(const ModInt &r) const { if (this->val < r.val) return ModInt(0); return this->fac() / ((*this - r).fac() * r.fac()); } ModInt sqrt() const { if (val == 0) return 0; if (mod == 2) return val; if (power((mod - 1) / 2) != 1) return 0; ModInt b = 1; while (b.power((mod - 1) / 2) == 1) b += 1; int e = 0, m = mod - 1; while (m % 2 == 0) m >>= 1, e++; ModInt x = power((m - 1) / 2), y = (*this) * x * x; x *= (*this); ModInt z = b.power(m); while (y != 1) { int j = 0; ModInt t = y; while (t != 1) j++, t *= t; z = z.power(1LL << (e - j - 1)); x *= z, z *= z, y *= z; e = j; } return ModInt(std::min(x.val, mod - x.val)); } }; using mint = ModInt; struct cmplx{ double x, y; cmplx() : x(0), y(0) {} cmplx(double x, double y) : x(x), y(y) {} inline cmplx operator+(const cmplx &r) const { return cmplx(x + r.x, y + r.y); } inline cmplx operator-(const cmplx &r) const { return cmplx(x - r.x, y - r.y); } inline cmplx operator*(const cmplx &r) const { return cmplx(x * r.x - y * r.y, x * r.y + y * r.x); } inline cmplx conj() const { return cmplx(x, -y); } }; int fftbase = 1; vector fftrts = {{0, 0}, {1, 0}}; vector fftrev = {0, 1}; void ensure_base(int nbase) { if (nbase <= fftbase) return; fftrev.resize(1 << nbase); fftrts.resize(1 << nbase); for (int i = 0; i < (1 << nbase); i++) { fftrev[i] = (fftrev[i >> 1] >> 1) + ((i & 1) << (nbase - 1)); } while (fftbase < nbase) { double angle = M_PI * 2.0 / (1 << (fftbase + 1)); for (int i = 1 << (fftbase - 1); i < (1 << fftbase); i++) { fftrts[i << 1] = fftrts[i]; double angle_i = angle * (2 * i + 1 - (1 << fftbase)); fftrts[(i << 1) + 1] = {cos(angle_i), sin(angle_i)}; } ++fftbase; } } void fft(int n, vector &a) { assert((n & (n - 1)) == 0); int zeros = __builtin_ctz(n); ensure_base(zeros); int shift = fftbase - zeros; for (int i = 0; i < n; i++) { if (i < (fftrev[i] >> shift)) { swap(a[i], a[fftrev[i] >> shift]); } } for (int k = 1; k < n; k <<= 1) { for (int i = 0; i < n; i += 2 * k) { for (int j = 0; j < k; j++) { cmplx z = a[i + j + k] * fftrts[j + k]; a[i + j + k] = a[i + j] - z; a[i + j] = a[i + j] + z; } } } } // Convolution for ModInt class // retval[i] = \sum_j a[j] b[i - j] template vector convolution_mod(vector a, vector b) { int need = int(a.size() + b.size()) - 1; int nbase = 0; while ((1 << nbase) < need) nbase++; int sz = 1 << nbase; vector fa(sz); for (int i = 0; i < (int)a.size(); i++) fa[i] = {double(a[i].val & ((1 << 15) - 1)), double(a[i].val >> 15)}; fft(sz, fa); vector fb(sz); if (a == b) fb = fa; else { for (int i = 0; i < (int)b.size(); i++) fb[i] = {double(b[i].val & ((1 << 15) - 1)), double(b[i].val >> 15)}; fft(sz, fb); } double ratio = 0.25 / sz; cmplx r2(0, -1), r3(ratio, 0), r4(0, -ratio), r5(0, 1); for (int i = 0; i <= (sz >> 1); i++) { int j = (sz - i) & (sz - 1); cmplx a1 = (fa[i] + fa[j].conj()); cmplx a2 = (fa[i] - fa[j].conj()) * r2; cmplx b1 = (fb[i] + fb[j].conj()) * r3; cmplx b2 = (fb[i] - fb[j].conj()) * r4; if (i != j) { cmplx c1 = (fa[j] + fa[i].conj()); cmplx c2 = (fa[j] - fa[i].conj()) * r2; cmplx d1 = (fb[j] + fb[i].conj()) * r3; cmplx d2 = (fb[j] - fb[i].conj()) * r4; fa[i] = c1 * d1 + c2 * d2 * r5; fb[i] = c1 * d2 + c2 * d1; } fa[j] = a1 * b1 + a2 * b2 * r5; fb[j] = a1 * b2 + a2 * b1; } fft(sz, fa); fft(sz, fb); vector ret(sz); for (int i = 0; i < need; i++) { int64_t aa = llround(fa[i].x); int64_t bb = llround(fb[i].x); int64_t cc = llround(fa[i].y); aa = MODINT(aa).val, bb = MODINT(bb).val, cc = MODINT(cc).val; ret[i] = aa + (bb << 15) + (cc << 30); } return ret; } template struct FormalPowerSeries : vector { using vector::vector; using P = FormalPowerSeries; void shrink() { while (this->size() and this->back() == T(0)) this->pop_back(); } P operator+(const P &r) const { return P(*this) += r; } P operator+(const T &v) const { return P(*this) += v; } P operator-(const P &r) const { return P(*this) -= r; } P operator-(const T &v) const { return P(*this) -= v; } P operator*(const P &r) const { return P(*this) *= r; } P operator*(const T &v) const { return P(*this) *= v; } P operator/(const P &r) const { return P(*this) /= r; } P operator/(const T &v) const { return P(*this) /= v; } P operator%(const P &r) const { return P(*this) %= r; } P &operator+=(const P &r) { if (r.size() > this->size()) this->resize(r.size()); for (int i = 0; i < (int)r.size(); i++) (*this)[i] += r[i]; shrink(); return *this; } P &operator+=(const T &v) { if (this->empty()) this->resize(1); (*this)[0] += v; shrink(); return *this; } P &operator-=(const P &r) { if(r.size() > this->size()) this->resize(r.size()); for(int i = 0; i < (int)r.size(); i++) (*this)[i] -= r[i]; shrink(); return *this; } P &operator-=(const T &v) { if(this->empty()) this->resize(1); (*this)[0] -= v; shrink(); return *this; } P &operator*=(const T &v) { for (auto &x : (*this)) x *= v; shrink(); return *this; } P &operator*=(const P &r) { if (this->empty() || r.empty()) this->clear(); else { auto ret = convolution_mod(*this, r); *this = P(ret.begin(), ret.end()); } return *this; } P &operator%=(const P &r) { *this -= *this / r * r; shrink(); return *this; } P operator-() const { P ret = *this; for (auto &v : ret) v = -v; return ret; } P &operator/=(const T &v) { assert(v != T(0)); for (auto &x : (*this)) x /= v; return *this; } P &operator/=(const P &r) { if (this->size() < r.size()) { this->clear(); return *this; } int n = (int)this->size() - r.size() + 1; return *this = (reversed().pre(n) * r.reversed().inv(n)).pre(n).reversed(n); } P pre(int sz) const { P ret(this->begin(), this->begin() + min((int)this->size(), sz)); ret.shrink(); return ret; } P operator>>(int sz) const { if ((int)this->size() <= sz) return {}; return P(this->begin() + sz, this->end()); } P operator<<(int sz) const { if (this->empty()) return {}; P ret(*this); ret.insert(ret.begin(), sz, T(0)); return ret; } P reversed(int deg = -1) const { assert(deg >= -1); P ret(*this); if (deg != -1) ret.resize(deg, T(0)); reverse(ret.begin(), ret.end()); ret.shrink(); return ret; } P differential() const { // formal derivative (differential) of f.p.s. const int n = (int)this->size(); P ret(max(0, n - 1)); for (int i = 1; i < n; i++) ret[i - 1] = (*this)[i] * T(i); return ret; } P integral() const { const int n = (int)this->size(); P ret(n + 1); ret[0] = T(0); for (int i = 0; i < n; i++) ret[i + 1] = (*this)[i] / T(i + 1); return ret; } P inv(int deg) const { assert(deg >= -1); assert(this->size() and ((*this)[0]) != T(0)); // Requirement: F(0) != 0 const int n = this->size(); if (deg == -1) deg = n; P ret({T(1) / (*this)[0]}); for (int i = 1; i < deg; i <<= 1) { ret = (ret + ret - ret * ret * pre(i << 1)).pre(i << 1); } ret = ret.pre(deg); ret.shrink(); return ret; } P log(int deg = -1) const { assert(deg >= -1); assert(this->size() and ((*this)[0]) == T(1)); // Requirement: F(0) = 1 const int n = (int)this->size(); if (deg == 0) return {}; if (deg == -1) deg = n; return (this->differential() * this->inv(deg)).pre(deg - 1).integral(); } P sqrt(int deg = -1) const { assert(deg >= -1); const int n = (int)this->size(); if (deg == -1) deg = n; if (this->empty()) return {}; if ((*this)[0] == T(0)) { for (int i = 1; i < n; i++) if ((*this)[i] != T(0)) { if ((i & 1) or deg - i / 2 <= 0) return {}; return (*this >> i).sqrt(deg - i / 2) << (i / 2); } return {}; } T sqrtf0 = (*this)[0].sqrt(); if (sqrtf0 == T(0)) return {}; P y = (*this) / (*this)[0], ret({T(1)}); T inv2 = T(1) / T(2); for (int i = 1; i < deg; i <<= 1) { ret = (ret + y.pre(i << 1) * ret.inv(i << 1)) * inv2; } return ret.pre(deg) * sqrtf0; } P exp(int deg = -1) const { assert(deg >= -1); assert(this->empty() or ((*this)[0]) == T(0)); // Requirement: F(0) = 0 const int n = (int)this->size(); if (deg == -1) deg = n; P ret({T(1)}); for (int i = 1; i < deg; i <<= 1) { ret = (ret * (pre(i << 1) + T(1) - ret.log(i << 1))).pre(i << 1); } return ret.pre(deg); } P pow(long long int k, int deg = -1) const { assert(deg >= -1); const int n = (int)this->size(); if (deg == -1) deg = n; for (int i = 0; i < n; i++) { if ((*this)[i] != T(0)) { T rev = T(1) / (*this)[i]; P C(*this * rev); P D(n - i); for (int j = i; j < n; j++) D[j - i] = C[j]; D = (D.log(deg) * T(k)).exp(deg) * (*this)[i].power(k); P E(deg); if (k * (i > 0) > deg or k * i > deg) return {}; long long int S = i * k; for (int j = 0; j + S < deg and j < (int)D.size(); j++) E[j + S] = D[j]; E.shrink(); return E; } } return *this; } T coeff(int i) const { if ((int)this->size() <= i) return T(0); return (*this)[i]; } T eval(T x) const { T ret = 0, w = 1; for (auto &v : *this) ret += w * v, w *= x; return ret; } }; int main() { int N; cin >> N; FormalPowerSeries cosx2(N * 2 + 1), sinx2(N * 2 + 1); REP(i, N) { cosx2.at(i * 2) = mint(1) / mint(i * 2).fac() / mint(2).power(i * 2) * (i % 2 ? -1 : 1); sinx2.at(i * 2 + 1) = mint(1) / mint(i * 2 + 1).fac() / mint(2).power(i * 2 + 1) * (i % 2 ? -1 : 1); } auto cos_plus_sin = cosx2 + sinx2; auto cos_minus_sin_inv = (cosx2 - sinx2).inv(N + 2); auto ret = cos_plus_sin * cos_minus_sin_inv; cout << ret.coeff(N) * 2 * mint(N).fac() << endl; }