#define _USE_MATH_DEFINES #include using namespace std; #define FOR(i,m,n) for(int i=(m);i<(n);++i) #define REP(i,n) FOR(i,0,n) #define ALL(v) (v).begin(),(v).end() using ll = long long; constexpr int INF = 0x3f3f3f3f; constexpr long long LINF = 0x3f3f3f3f3f3f3f3fLL; constexpr double EPS = 1e-8; constexpr int MOD = 998244353; // constexpr int MOD = 1000000007; constexpr int DY4[]{1, 0, -1, 0}, DX4[]{0, -1, 0, 1}; constexpr int DY8[]{1, 1, 0, -1, -1, -1, 0, 1}; constexpr int DX8[]{0, -1, -1, -1, 0, 1, 1, 1}; template inline bool chmax(T& a, U b) { return a < b ? (a = b, true) : false; } template inline bool chmin(T& a, U b) { return a > b ? (a = b, true) : false; } struct IOSetup { IOSetup() { std::cin.tie(nullptr); std::ios_base::sync_with_stdio(false); std::cout << fixed << setprecision(20); } } iosetup; template struct MInt { unsigned int v; MInt() : v(0) {} MInt(const long long x) : v(x >= 0 ? x % M : x % M + M) {} static constexpr int get_mod() { return M; } static void set_mod(const int divisor) { assert(divisor == M); } static void init(const int x = 10000000) { inv(x, true); fact(x); fact_inv(x); } static MInt inv(const int n, const bool init = false) { // assert(0 <= n && n < M && std::__gcd(n, M) == 1); static std::vector inverse{0, 1}; const int prev = inverse.size(); if (n < prev) { return inverse[n]; } else if (init) { // "n!" and "M" must be disjoint. inverse.resize(n + 1); for (int i = prev; i <= n; ++i) { inverse[i] = -inverse[M % i] * (M / i); } return inverse[n]; } int u = 1, v = 0; for (unsigned int a = n, b = M; b;) { const unsigned int q = a / b; std::swap(a -= q * b, b); std::swap(u -= q * v, v); } return u; } static MInt fact(const int n) { static std::vector factorial{1}; const int prev = factorial.size(); if (n >= prev) { factorial.resize(n + 1); for (int i = prev; i <= n; ++i) { factorial[i] = factorial[i - 1] * i; } } return factorial[n]; } static MInt fact_inv(const int n) { static std::vector f_inv{1}; const int prev = f_inv.size(); if (n >= prev) { f_inv.resize(n + 1); f_inv[n] = inv(fact(n).v); for (int i = n; i > prev; --i) { f_inv[i - 1] = f_inv[i] * i; } } return f_inv[n]; } static MInt nCk(const int n, const int k) { if (n < 0 || n < k || k < 0) return 0; return fact(n) * (n - k < k ? fact_inv(k) * fact_inv(n - k) : fact_inv(n - k) * fact_inv(k)); } static MInt nPk(const int n, const int k) { return n < 0 || n < k || k < 0 ? 0 : fact(n) * fact_inv(n - k); } static MInt nHk(const int n, const int k) { return n < 0 || k < 0 ? 0 : (k == 0 ? 1 : nCk(n + k - 1, k)); } static MInt large_nCk(long long n, const int k) { if (n < 0 || n < k || k < 0) return 0; inv(k, true); MInt res = 1; for (int i = 1; i <= k; ++i) { res *= inv(i) * n--; } return res; } MInt pow(long long exponent) const { MInt res = 1, tmp = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) res *= tmp; tmp *= tmp; } return res; } MInt& operator+=(const MInt& x) { if ((v += x.v) >= M) v -= M; return *this; } MInt& operator-=(const MInt& x) { if ((v += M - x.v) >= M) v -= M; return *this; } MInt& operator*=(const MInt& x) { v = static_cast(v) * x.v % M; return *this; } MInt& operator/=(const MInt& x) { return *this *= inv(x.v); } bool operator==(const MInt& x) const { return v == x.v; } bool operator!=(const MInt& x) const { return v != x.v; } bool operator<(const MInt& x) const { return v < x.v; } bool operator<=(const MInt& x) const { return v <= x.v; } bool operator>(const MInt& x) const { return v > x.v; } bool operator>=(const MInt& x) const { return v >= x.v; } MInt& operator++() { if (++v == M) v = 0; return *this; } MInt operator++(int) { const MInt res = *this; ++*this; return res; } MInt& operator--() { v = (v == 0 ? M - 1 : v - 1); return *this; } MInt operator--(int) { const MInt res = *this; --*this; return res; } MInt operator+() const { return *this; } MInt operator-() const { return MInt(v ? M - v : 0); } MInt operator+(const MInt& x) const { return MInt(*this) += x; } MInt operator-(const MInt& x) const { return MInt(*this) -= x; } MInt operator*(const MInt& x) const { return MInt(*this) *= x; } MInt operator/(const MInt& x) const { return MInt(*this) /= x; } friend std::ostream& operator<<(std::ostream& os, const MInt& x) { return os << x.v; } friend std::istream& operator>>(std::istream& is, MInt& x) { long long v; is >> v; x = MInt(v); return is; } }; using ModInt = MInt; template struct NumberTheoreticTransform { using ModInt = MInt; NumberTheoreticTransform() { for (int i = 0; i < 23; ++i) { if (primes[i][0] == ModInt::get_mod()) { n_max = 1 << primes[i][2]; root = ModInt(primes[i][1]).pow((primes[i][0] - 1) >> primes[i][2]); return; } } assert(false); } template std::vector dft(const std::vector& a) { const int n = a.size(); int lg = 1; while ((1 << lg) < n) ++lg; std::vector b(1 << lg, 0); std::copy(a.begin(), a.end(), b.begin()); calc(&b); return b; } void idft(std::vector* a) { const int n = a->size(); assert(__builtin_popcount(n) == 1); calc(a); std::reverse(std::next(a->begin()), a->end()); const ModInt inv_n = ModInt::inv(n); for (int i = 0; i < n; ++i) { (*a)[i] *= inv_n; } } template std::vector convolution(const std::vector& a, const std::vector& b) { const int a_size = a.size(), b_size = b.size(); const int c_size = a_size + b_size - 1; int lg = 1; while ((1 << lg) < c_size) ++lg; const int n = 1 << lg; std::vector c(n, 0), d(n, 0); std::copy(a.begin(), a.end(), c.begin()); calc(&c); std::copy(b.begin(), b.end(), d.begin()); calc(&d); for (int i = 0; i < n; ++i) { c[i] *= d[i]; } idft(&c); c.resize(c_size); return c; } private: const int primes[23][3]{ {16957441, 329, 14}, {17006593, 26, 15}, {19529729, 770, 17}, {167772161, 3, 25}, {469762049, 3, 26}, {645922817, 3, 23}, {897581057, 3, 23}, {924844033, 5, 21}, {935329793, 3, 22}, {943718401, 7, 22}, {950009857, 7, 21}, {962592769, 7, 21}, {975175681, 17, 21}, {976224257, 3, 20}, {985661441, 3, 22}, {998244353, 3, 23}, {1004535809, 3, 21}, {1007681537, 3, 20}, {1012924417, 5, 21}, {1045430273, 3, 20}, {1051721729, 6, 20}, {1053818881, 7, 20}, {1224736769, 3, 24} }; int n_max; ModInt root; std::vector butterfly{0}; std::vector> omega{{1}}; void calc(std::vector* a) { const int n = a->size(), prev_n = butterfly.size(); if (n > prev_n) { assert(n <= n_max); butterfly.resize(n); const int prev_lg = omega.size(), lg = __builtin_ctz(n); for (int i = 1; i < prev_n; ++i) { butterfly[i] <<= lg - prev_lg; } for (int i = prev_n; i < n; ++i) { butterfly[i] = (butterfly[i >> 1] >> 1) | ((i & 1) << (lg - 1)); } omega.resize(lg); for (int i = prev_lg; i < lg; ++i) { omega[i].resize(1 << i); const ModInt tmp = root.pow((ModInt::get_mod() - 1) >> (i + 1)); for (int j = 0; j < (1 << (i - 1)); ++j) { omega[i][j << 1] = omega[i - 1][j]; omega[i][(j << 1) + 1] = omega[i - 1][j] * tmp; } } } const int shift = __builtin_ctz(butterfly.size()) - __builtin_ctz(n); for (int i = 0; i < n; ++i) { const int j = butterfly[i] >> shift; if (i < j) std::swap((*a)[i], (*a)[j]); } for (int block = 1, den = 0; block < n; block <<= 1, ++den) { for (int i = 0; i < n; i += (block << 1)) { for (int j = 0; j < block; ++j) { const ModInt tmp = (*a)[i + j + block] * omega[den][j]; (*a)[i + j + block] = (*a)[i + j] - tmp; (*a)[i + j] += tmp; } } } } }; template struct FormalPowerSeries { std::vector coef; explicit FormalPowerSeries(const int deg = 0) : coef(deg + 1, 0) {} explicit FormalPowerSeries(const std::vector& coef) : coef(coef) {} FormalPowerSeries(const std::initializer_list init) : coef(init.begin(), init.end()) {} template explicit FormalPowerSeries(const InputIter first, const InputIter last) : coef(first, last) {} inline const T& operator[](const int term) const { return coef[term]; } inline T& operator[](const int term) { return coef[term]; } using Mult = std::function(const std::vector&, const std::vector&)>; using Sqrt = std::function; static void set_mult(const Mult mult) { get_mult() = mult; } static void set_sqrt(const Sqrt sqrt) { get_sqrt() = sqrt; } void resize(const int deg) { coef.resize(deg + 1, 0); } void shrink() { while (coef.size() > 1 && coef.back() == 0) coef.pop_back(); } int degree() const { return static_cast(coef.size()) - 1; } FormalPowerSeries& operator=(const std::vector& coef_) { coef = coef_; return *this; } FormalPowerSeries& operator=(const FormalPowerSeries& x) = default; FormalPowerSeries& operator+=(const FormalPowerSeries& x) { const int deg_x = x.degree(); if (deg_x > degree()) resize(deg_x); for (int i = 0; i <= deg_x; ++i) { coef[i] += x[i]; } return *this; } FormalPowerSeries& operator-=(const FormalPowerSeries& x) { const int deg_x = x.degree(); if (deg_x > degree()) resize(deg_x); for (int i = 0; i <= deg_x; ++i) { coef[i] -= x[i]; } return *this; } FormalPowerSeries& operator*=(const T x) { for (T& e : coef) e *= x; return *this; } FormalPowerSeries& operator*=(const FormalPowerSeries& x) { return *this = get_mult()(coef, x.coef); } FormalPowerSeries& operator/=(const T x) { assert(x != 0); return *this *= static_cast(1) / x; } FormalPowerSeries& operator/=(const FormalPowerSeries& x) { const int n = degree() - x.degree() + 1; if (n <= 0) return *this = FormalPowerSeries(); const std::vector tmp = get_mult()( std::vector(coef.rbegin(), std::next(coef.rbegin(), n)), FormalPowerSeries( x.coef.rbegin(), std::next(x.coef.rbegin(), std::min(x.degree() + 1, n))) .inv(n - 1).coef); return *this = FormalPowerSeries(std::prev(tmp.rend(), n), tmp.rend()); } FormalPowerSeries& operator%=(const FormalPowerSeries& x) { if (x.degree() == 0) return *this = FormalPowerSeries{0}; *this -= *this / x * x; resize(x.degree() - 1); return *this; } FormalPowerSeries& operator<<=(const int n) { coef.insert(coef.begin(), n, 0); return *this; } FormalPowerSeries& operator>>=(const int n) { if (degree() < n) return *this = FormalPowerSeries(); coef.erase(coef.begin(), coef.begin() + n); return *this; } bool operator==(FormalPowerSeries x) const { x.shrink(); FormalPowerSeries y = *this; y.shrink(); return x.coef == y.coef; } bool operator!=(const FormalPowerSeries& x) const { return !(*this == x); } FormalPowerSeries operator+() const { return *this; } FormalPowerSeries operator-() const { FormalPowerSeries res = *this; for (T& e : res.coef) e = -e; return res; } FormalPowerSeries operator+(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) += x; } FormalPowerSeries operator-(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) -= x; } FormalPowerSeries operator*(const T x) const { return FormalPowerSeries(*this) *= x; } FormalPowerSeries operator*(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) *= x; } FormalPowerSeries operator/(const T x) const { return FormalPowerSeries(*this) /= x; } FormalPowerSeries operator/(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) /= x; } FormalPowerSeries operator%(const FormalPowerSeries& x) const { return FormalPowerSeries(*this) %= x; } FormalPowerSeries operator<<(const int n) const { return FormalPowerSeries(*this) <<= n; } FormalPowerSeries operator>>(const int n) const { return FormalPowerSeries(*this) >>= n; } T horner(const T x) const { T res = 0; for (int i = degree(); i >= 0; --i) { res = res * x + coef[i]; } return res; } FormalPowerSeries differential() const { const int deg = degree(); assert(deg >= 0); FormalPowerSeries res(std::max(deg - 1, 0)); for (int i = 1; i <= deg; ++i) { res[i - 1] = coef[i] * i; } return res; } FormalPowerSeries exp(int deg = -1) const { assert(coef[0] == 0); const int n = coef.size(); if (deg == -1) deg = n - 1; const FormalPowerSeries one{1}; FormalPowerSeries res = one; for (int i = 1; i <= deg; i <<= 1) { res *= FormalPowerSeries(coef.begin(), std::next(coef.begin(), std::min(n, i << 1))) - res.log((i << 1) - 1) + one; res.coef.resize(i << 1); } res.resize(deg); return res; } FormalPowerSeries inv(int deg = -1) const { assert(coef[0] != 0); const int n = coef.size(); if (deg == -1) deg = n - 1; FormalPowerSeries res{static_cast(1) / coef[0]}; for (int i = 1; i <= deg; i <<= 1) { res = res + res - res * res * FormalPowerSeries( coef.begin(), std::next(coef.begin(), std::min(n, i << 1))); res.coef.resize(i << 1); } res.resize(deg); return res; } FormalPowerSeries log(int deg = -1) const { assert(coef[0] == 1); if (deg == -1) deg = degree(); FormalPowerSeries integrand = differential() * inv(deg - 1); integrand.resize(deg); for (int i = deg; i > 0; --i) { integrand[i] = integrand[i - 1] / i; } integrand[0] = 0; return integrand; } FormalPowerSeries pow(long long exponent, int deg = -1) const { const int n = coef.size(); if (deg == -1) deg = n - 1; if (exponent == 0) { FormalPowerSeries res(deg); if (deg != -1) res[0] = 1; return res; } assert(deg >= 0); for (int i = 0; i < n; ++i) { if (coef[i] == 0) continue; if (i > deg / exponent) break; const long long shift = exponent * i; T tmp = 1, base = coef[i]; for (long long e = exponent; e > 0; e >>= 1) { if (e & 1) tmp *= base; base *= base; } const FormalPowerSeries res = ((*this >> i) / coef[i]).log(deg - shift); return ((res * exponent).exp(deg - shift) * tmp) << shift; } return FormalPowerSeries(deg); } FormalPowerSeries mod_pow(long long exponent, const FormalPowerSeries& md) const { const int deg = md.degree() - 1; if (deg < 0) return FormalPowerSeries(-1); const FormalPowerSeries inv_rev_md = FormalPowerSeries(md.coef.rbegin(), md.coef.rend()).inv(); const auto mod_mult = [&md, &inv_rev_md, deg]( FormalPowerSeries* multiplicand, const FormalPowerSeries& multiplier) -> void { *multiplicand *= multiplier; if (deg < multiplicand->degree()) { const int n = multiplicand->degree() - deg; const FormalPowerSeries quotient = FormalPowerSeries(multiplicand->coef.rbegin(), std::next(multiplicand->coef.rbegin(), n)) * FormalPowerSeries( inv_rev_md.coef.begin(), std::next(inv_rev_md.coef.begin(), std::min(deg + 2, n))); *multiplicand -= FormalPowerSeries(std::prev(quotient.coef.rend(), n), quotient.coef.rend()) * md; multiplicand->resize(deg); } multiplicand->shrink(); }; FormalPowerSeries res{1}, base = *this; for (; exponent > 0; exponent >>= 1) { if (exponent & 1) mod_mult(&res, base); mod_mult(&base, base); } return res; } FormalPowerSeries sqrt(int deg = -1) const { const int n = coef.size(); if (deg == -1) deg = n - 1; if (coef[0] == 0) { for (int i = 1; i < n; ++i) { if (coef[i] == 0) continue; if (i & 1) return FormalPowerSeries(-1); const int shift = i >> 1; if (deg < shift) break; FormalPowerSeries res = (*this >> i).sqrt(deg - shift); if (res.coef.empty()) return FormalPowerSeries(-1); res <<= shift; res.resize(deg); return res; } return FormalPowerSeries(deg); } T s; if (!get_sqrt()(coef.front(), &s)) return FormalPowerSeries(-1); FormalPowerSeries res{s}; const T half = static_cast(1) / 2; for (int i = 1; i <= deg; i <<= 1) { res = (FormalPowerSeries(coef.begin(), std::next(coef.begin(), std::min(n, i << 1))) * res.inv((i << 1) - 1) + res) * half; } res.resize(deg); return res; } FormalPowerSeries translate(const T c) const { const int n = coef.size(); std::vector fact(n, 1), inv_fact(n, 1); for (int i = 1; i < n; ++i) { fact[i] = fact[i - 1] * i; } inv_fact[n - 1] = static_cast(1) / fact[n - 1]; for (int i = n - 1; i > 0; --i) { inv_fact[i - 1] = inv_fact[i] * i; } std::vector g(n), ex(n); for (int i = 0; i < n; ++i) { g[i] = coef[i] * fact[i]; } std::reverse(g.begin(), g.end()); T pow_c = 1; for (int i = 0; i < n; ++i) { ex[i] = pow_c * inv_fact[i]; pow_c *= c; } const std::vector conv = get_mult()(g, ex); FormalPowerSeries res(n - 1); for (int i = 0; i < n; ++i) { res[i] = conv[n - 1 - i] * inv_fact[i]; } return res; } private: static Mult& get_mult() { static Mult mult = [](const std::vector& a, const std::vector& b) -> std::vector { const int n = a.size(), m = b.size(); std::vector res(n + m - 1, 0); for (int i = 0; i < n; ++i) { for (int j = 0; j < m; ++j) { res[i + j] += a[i] * b[j]; } } return res; }; return mult; } static Sqrt& get_sqrt() { static Sqrt sqrt = [](const T& a, T* res) -> bool { return false; }; return sqrt; } }; template std::vector subset_sum_problem(const std::vector& a, const int d) { T zero = 1; std::vector nums(d + 1, 0); for (const int e : a) { if (e == 0) { zero *= 2; } else if (e <= d) { ++nums[e]; } } FormalPowerSeries fps(d); for (int i = 1; i <= d; ++i) { for (int j = 1; i * j <= d; ++j) { fps[i * j] += static_cast(nums[i]) / (j & 1 ? j : -j); } } return (fps.exp(d) * zero).coef; } int main() { constexpr int P = 999630629; int n; cin >> n; vector a(n); REP(i, n) cin >> a[i]; const int s = accumulate(ALL(a), 0); ModInt ans = ModInt(2).pow(n - 1) * s; if (s >= P) { const vector sum = subset_sum_problem(a, s - P); ans -= accumulate(ALL(sum), ModInt(0)) * P; } cout << ans << '\n'; return 0; }