// #pragma GCC optimize("O3,unroll-loops") #include // #include using namespace std; #if __cplusplus >= 202002L using namespace numbers; #endif template struct modular_fixed_base{ #define IS_INTEGRAL(T) (is_integral_v || is_same_v || is_same_v) #define IS_UNSIGNED(T) (is_unsigned_v || is_same_v) static_assert(IS_UNSIGNED(data_t)); static_assert(_mod >= 1); static constexpr bool VARIATE_MOD_FLAG = false; static constexpr data_t mod(){ return _mod; } template static vector precalc_power(T base, int SZ){ vector res(SZ + 1, 1); for(auto i = 1; i <= SZ; ++ i) res[i] = res[i - 1] * base; return res; } static vector _INV; static void precalc_inverse(int SZ){ if(_INV.empty()) _INV.assign(2, 1); for(auto x = _INV.size(); x <= SZ; ++ x) _INV.push_back(_mod / x * -_INV[_mod % x]); } // _mod must be a prime static modular_fixed_base _primitive_root; static modular_fixed_base primitive_root(){ if(_primitive_root) return _primitive_root; if(_mod == 2) return _primitive_root = 1; if(_mod == 998244353) return _primitive_root = 3; data_t divs[20] = {}; divs[0] = 2; int cnt = 1; data_t x = (_mod - 1) / 2; while(x % 2 == 0) x /= 2; for(auto i = 3; 1LL * i * i <= x; i += 2){ if(x % i == 0){ divs[cnt ++] = i; while(x % i == 0) x /= i; } } if(x > 1) divs[cnt ++] = x; for(auto g = 2; ; ++ g){ bool ok = true; for(auto i = 0; i < cnt; ++ i){ if(modular_fixed_base(g).power((_mod - 1) / divs[i]) == 1){ ok = false; break; } } if(ok) return _primitive_root = g; } } constexpr modular_fixed_base(){ } modular_fixed_base(const double &x){ data = _normalize(llround(x)); } modular_fixed_base(const long double &x){ data = _normalize(llround(x)); } template::type* = nullptr> modular_fixed_base(const T &x){ data = _normalize(x); } template::type* = nullptr> static data_t _normalize(const T &x){ int sign = x >= 0 ? 1 : -1; data_t v = _mod <= sign * x ? sign * x % _mod : sign * x; if(sign == -1 && v) v = _mod - v; return v; } template::type* = nullptr> operator T() const{ return data; } modular_fixed_base &operator+=(const modular_fixed_base &otr){ if((data += otr.data) >= _mod) data -= _mod; return *this; } modular_fixed_base &operator-=(const modular_fixed_base &otr){ if((data += _mod - otr.data) >= _mod) data -= _mod; return *this; } template::type* = nullptr> modular_fixed_base &operator+=(const T &otr){ return *this += modular_fixed_base(otr); } template::type* = nullptr> modular_fixed_base &operator-=(const T &otr){ return *this -= modular_fixed_base(otr); } modular_fixed_base &operator++(){ return *this += 1; } modular_fixed_base &operator--(){ return *this += _mod - 1; } modular_fixed_base operator++(int){ modular_fixed_base result(*this); *this += 1; return result; } modular_fixed_base operator--(int){ modular_fixed_base result(*this); *this += _mod - 1; return result; } modular_fixed_base operator-() const{ return modular_fixed_base(_mod - data); } modular_fixed_base &operator*=(const modular_fixed_base &rhs){ if constexpr(is_same_v) data = (unsigned long long)data * rhs.data % _mod; else if constexpr(is_same_v){ long long res = data * rhs.data - _mod * (unsigned long long)(1.L / _mod * data * rhs.data); data = res + _mod * (res < 0) - _mod * (res >= (long long)_mod); } else data = _normalize(data * rhs.data); return *this; } template::type* = nullptr> modular_fixed_base &inplace_power(T e){ if(e == 0) return *this = 1; if(data == 0) return *this = {}; if(data == 1 || e == 1) return *this; if(data == mod() - 1) return e % 2 ? *this : *this = -*this; if(e < 0) *this = 1 / *this, e = -e; if(e == 1) return *this; modular_fixed_base res = 1; for(; e; *this *= *this, e >>= 1) if(e & 1) res *= *this; return *this = res; } template::type* = nullptr> modular_fixed_base power(T e) const{ return modular_fixed_base(*this).inplace_power(e); } modular_fixed_base &operator/=(const modular_fixed_base &otr){ make_signed_t a = otr.data, m = _mod, u = 0, v = 1; if(a < _INV.size()) return *this *= _INV[a]; while(a){ make_signed_t t = m / a; m -= t * a; swap(a, m); u -= t * v; swap(u, v); } assert(m == 1); return *this *= u; } #define ARITHMETIC_OP(op, apply_op)\ modular_fixed_base operator op(const modular_fixed_base &x) const{ return modular_fixed_base(*this) apply_op x; }\ template::type* = nullptr>\ modular_fixed_base operator op(const T &x) const{ return modular_fixed_base(*this) apply_op modular_fixed_base(x); }\ template::type* = nullptr>\ friend modular_fixed_base operator op(const T &x, const modular_fixed_base &y){ return modular_fixed_base(x) apply_op y; } ARITHMETIC_OP(+, +=) ARITHMETIC_OP(-, -=) ARITHMETIC_OP(*, *=) ARITHMETIC_OP(/, /=) #undef ARITHMETIC_OP #define COMPARE_OP(op)\ bool operator op(const modular_fixed_base &x) const{ return data op x.data; }\ template::type* = nullptr>\ bool operator op(const T &x) const{ return data op modular_fixed_base(x).data; }\ template::type* = nullptr>\ friend bool operator op(const T &x, const modular_fixed_base &y){ return modular_fixed_base(x).data op y.data; } COMPARE_OP(==) COMPARE_OP(!=) COMPARE_OP(<) COMPARE_OP(<=) COMPARE_OP(>) COMPARE_OP(>=) #undef COMPARE_OP friend istream &operator>>(istream &in, modular_fixed_base &number){ long long x; in >> x; number.data = modular_fixed_base::_normalize(x); return in; } //#define _SHOW_FRACTION friend ostream &operator<<(ostream &out, const modular_fixed_base &number){ out << number.data; #if defined(LOCAL) && defined(_SHOW_FRACTION) cerr << "("; for(auto d = 1; ; ++ d){ if((number * d).data <= 1000000){ cerr << (number * d).data; if(d != 1) cerr << "/" << d; break; } else if((-number * d).data <= 1000000){ cerr << "-" << (-number * d).data; if(d != 1) cerr << "/" << d; break; } } cerr << ")"; #endif return out; } data_t data = 0; #undef _SHOW_FRACTION #undef IS_INTEGRAL #undef IS_UNSIGNED }; template vector> modular_fixed_base::_INV; template modular_fixed_base modular_fixed_base::_primitive_root; const unsigned int mod = (119 << 23) + 1; // 998244353 // const unsigned int mod = 1e9 + 7; // 1000000007 // const unsigned int mod = 1e9 + 9; // 1000000009 // const unsigned long long mod = (unsigned long long)1e18 + 9; using modular = modular_fixed_base, mod>; template struct combinatorics{ #ifdef LOCAL #define ASSERT(c) assert(c) #else #define ASSERT(c) 42 #endif // O(n) static vector precalc_fact(int n){ vector f(n + 1, 1); for(auto i = 1; i <= n; ++ i) f[i] = f[i - 1] * i; return f; } // O(n * m) static vector> precalc_C(int n, int m){ vector> c(n + 1, vector(m + 1)); for(auto i = 0; i <= n; ++ i) for(auto j = 0; j <= min(i, m); ++ j) c[i][j] = i && j ? c[i - 1][j - 1] + c[i - 1][j] : T(1); return c; } int SZ = 0; vector inv, fact, invfact; combinatorics(){ } // O(SZ) combinatorics(int SZ): SZ(SZ), inv(SZ + 1, 1), fact(SZ + 1, 1), invfact(SZ + 1, 1){ for(auto i = 1; i <= SZ; ++ i) fact[i] = fact[i - 1] * i; invfact[SZ] = 1 / fact[SZ]; for(auto i = SZ - 1; i >= 0; -- i){ invfact[i] = invfact[i + 1] * (i + 1); inv[i + 1] = invfact[i + 1] * fact[i]; } } // O(1) T C(int n, int k) const{ ASSERT(0 <= min(n, k) && max(n, k) <= SZ); return n >= k ? fact[n] * invfact[k] * invfact[n - k] : T{0}; } // O(1) T P(int n, int k) const{ ASSERT(0 <= min(n, k) && max(n, k) <= SZ); return n >= k ? fact[n] * invfact[n - k] : T{0}; } // O(1) T H(int n, int k) const{ ASSERT(0 <= min(n, k)); if(n == 0) return 0; return C(n + k - 1, k); } // Multinomial Coefficient T mC(int n, const vector &a) const{ ASSERT((int)a.size() >= 2 && accumulate(a.begin(), a.end(), 0) == n); ASSERT(0 <= min(n, *min_element(a.begin(), a.end())) && max(n, *max_element(a.begin(), a.end())) <= SZ); T res = fact[n]; for(auto x: a) res *= invfact[x]; return res; } // Multinomial Coefficient template && ...)>::type* = nullptr> T mC(int n, U... pack){ ASSERT(sizeof...(pack) >= 2 && (... + pack) == n); return (fact[n] * ... * invfact[pack]); } // O(min(k, n - k)) T naive_C(long long n, long long k) const{ ASSERT(0 <= min(n, k)); if(n < k) return 0; T res = 1; k = min(k, n - k); ASSERT(k <= SZ); for(auto i = n; i > n - k; -- i) res *= i; return res * invfact[k]; } // O(k) T naive_P(long long n, int k) const{ ASSERT(0 <= min(n, k)); if(n < k) return 0; T res = 1; for(auto i = n; i > n - k; -- i) res *= i; return res; } // O(k) T naive_H(long long n, int k) const{ ASSERT(0 <= min(n, k)); return naive_C(n + k - 1, k); } // O(1) bool parity_C(long long n, long long k) const{ ASSERT(0 <= min(n, k)); return n >= k ? (n & k) == k : false; } // Number of ways to place n '('s and k ')'s starting with m copies of '(' such that in each prefix, number of '(' is equal or greater than ')' // Catalan(n, n, 0): n-th catalan number // Catalan(s, s+k-1, k-1): sum of products of k catalan numbers where the index of product sums up to s. // O(1) T Catalan(int n, int k, int m = 0) const{ ASSERT(0 <= min({n, k, m})); return k <= m ? C(n + k, k) : k <= n + m ? C(n + k, k) - C(n + k, k - m - 1) : T(); } #undef ASSERT }; int main(){ cin.tie(0)->sync_with_stdio(0); cin.exceptions(ios::badbit | ios::failbit); int n; cin >> n; vector a(n); copy_n(istream_iterator(cin), n, a.begin()); if(n == 1){ cout << a[0] << "\n"; return 0; } combinatorics C(n << 1); modular res = 0; for(auto i = 0; i < n; ++ i){ res += a[i] * C.Catalan(n - 1 - i, n - 2, min(i, n - 2)); } cout << res << "\n"; return 0; } /* (1,0,0,0,0), (0,1,0,0,0), (0,0,1,0,0), (0,0,0,1,0), (0,0,0,0,1) (1,1,0,0,0), (1,1,1,0,0), (1,1,1,1,0), (1,1,1,1,1) (2,2,1,0,0), (3,3,2,1,0), (4,4,3,2,1) (5,5,3,1,0), (9,9,6,3,1) */