#include #include using namespace std; using namespace atcoder; //using mint = modint1000000007; //const int mod = 1000000007; using mint = modint998244353; const int mod = 998244353; //const int INF = 1e9; //const long long LINF = 1e18; #define rep(i, n) for (int i = 0; i < (n); ++i) #define rep2(i,l,r)for(int i=(l);i<(r);++i) #define rrep(i, n) for (int i = (n-1); i >= 0; --i) #define rrep2(i,l,r)for(int i=(r-1);i>=(l);--i) #define all(x) (x).begin(),(x).end() #define allR(x) (x).rbegin(),(x).rend() #define endl "\n" #define P pair template inline bool chmax(A & a, const B & b) { if (a < b) { a = b; return true; } return false; } template inline bool chmin(A & a, const B & b) { if (a > b) { a = b; return true; } return false; } // https://opt-cp.com/fps-implementation/ // verified by: // https://judge.yosupo.jp/problem/convolution_mod // https://judge.yosupo.jp/problem/inv_of_formal_power_series // https://judge.yosupo.jp/problem/log_of_formal_power_series // https://judge.yosupo.jp/problem/exp_of_formal_power_series // https://judge.yosupo.jp/problem/pow_of_formal_power_series // https://judge.yosupo.jp/problem/polynomial_taylor_shift // https://judge.yosupo.jp/problem/bernoulong longi_number // https://judge.yosupo.jp/problem/sharp_p_subset_sum //using mint = modint998244353; template struct Factorial { int MAX; vector fac, finv; Factorial(int m = 0) : MAX(m), fac(m + 1, 1), finv(m + 1, 1) { rep2(i, 2, MAX + 1) fac[i] = fac[i - 1] * i; finv[MAX] /= fac[MAX]; rrep2(i, 3, MAX + 1) finv[i - 1] = finv[i] * i; } T binom(int n, int k) { if (k < 0 || n < k) return 0; return fac[n] * finv[k] * finv[n - k]; } T perm(int n, int k) { if (k < 0 || n < k) return 0; return fac[n] * finv[n - k]; } }; Factorial fc; istream &operator>>(istream &is, modint998244353 &a) { long long v; is >> v; a = v; return is; } ostream &operator<<(ostream &os, const modint998244353 &a) { return os << a.val(); } istream &operator>>(istream &is, modint1000000007 &a) { long long v; is >> v; a = v; return is; } ostream &operator<<(ostream &os, const modint1000000007 &a) { return os << a.val(); } template istream &operator>>(istream &is, static_modint &a) { long long v; is >> v; a = v; return is; } template istream &operator>>(istream &is, dynamic_modint &a) { long long v; is >> v; a = v; return is; } template ostream &operator<<(ostream &os, const static_modint &a) { return os << a.val(); } template ostream &operator<<(ostream &os, const dynamic_modint &a) { return os << a.val(); } template istream &operator>>(istream &is, vector &v) { for (auto &e : v) is >> e; return is; } template ostream &operator<<(ostream &os, const vector &v) { for (auto &e : v) os << e << ' '; return os; } template struct FormalPowerSeries : vector { using vector::vector; using vector::operator=; using F = FormalPowerSeries; F operator-() const { F res(*this); for (auto &e : res) e = -e; return res; } F &operator*=(const T &g) { for (auto &e : *this) e *= g; return *this; } F &operator/=(const T &g) { assert(g != T(0)); *this *= g.inv(); return *this; } F &operator+=(const F &g) { int n = this->size(), m = g.size(); rep(i, min(n, m)) (*this)[i] += g[i]; return *this; } F &operator-=(const F &g) { int n = this->size(), m = g.size(); rep(i, min(n, m)) (*this)[i] -= g[i]; return *this; } F &operator<<=(const int d) { int n = this->size(); if (d >= n) *this = F(n); this->insert(this->begin(), d, 0); this->resize(n); return *this; } F &operator>>=(const int d) { int n = this->size(); this->erase(this->begin(), this->begin() + min(n, d)); this->resize(n); return *this; } // O(n log n) F inv(int d = -1) const { int n = this->size(); assert(n != 0 && (*this)[0] != 0); if (d == -1) d = n; assert(d >= 0); F res{ (*this)[0].inv() }; for (int m = 1; m < d; m *= 2) { F f(this->begin(), this->begin() + min(n, 2 * m)); F g(res); f.resize(2 * m), internal::butterfly(f); g.resize(2 * m), internal::butterfly(g); rep(i, 2 * m) f[i] *= g[i]; internal::butterfly_inv(f); f.erase(f.begin(), f.begin() + m); f.resize(2 * m), internal::butterfly(f); rep(i, 2 * m) f[i] *= g[i]; internal::butterfly_inv(f); T iz = T(2 * m).inv(); iz *= -iz; rep(i, m) f[i] *= iz; res.insert(res.end(), f.begin(), f.begin() + m); } res.resize(d); return res; } // fast: FMT-friendly modulus only // O(n log n) F &multiply_inplace(const F &g, int d = -1) { int n = this->size(); if (d == -1) d = n; assert(d >= 0); *this = convolution(move(*this), g); this->resize(d); return *this; } F multiply(const F &g, const int d = -1) const { return F(*this).multiply_inplace(g, d); } // O(n log n) F ÷_inplace(const F &g, int d = -1) { int n = this->size(); if (d == -1) d = n; assert(d >= 0); *this = convolution(move(*this), g.inv(d)); this->resize(d); return *this; } F divide(const F &g, const int d = -1) const { return F(*this).divide_inplace(g, d); } // // naive // // O(n^2) // F &multiply_inplace(const F &g) { // int n = this->size(), m = g.size(); // rrep(i, n) { // (*this)[i] *= g[0]; // rep2(j, 1, min(i+1, m)) (*this)[i] += (*this)[i-j] * g[j]; // } // return *this; // } // F multiply(const F &g) const { return F(*this).multiply_inplace(g); } // // O(n^2) // F ÷_inplace(const F &g) { // assert(g[0] != T(0)); // T ig0 = g[0].inv(); // int n = this->size(), m = g.size(); // rep(i, n) { // rep2(j, 1, min(i+1, m)) (*this)[i] -= (*this)[i-j] * g[j]; // (*this)[i] *= ig0; // } // return *this; // } // F divide(const F &g) const { return F(*this).divide_inplace(g); } // sparse // O(nk) F &multiply_inplace(vector> g) { int n = this->size(); auto[d, c] = g.front(); if (d == 0) g.erase(g.begin()); else c = 0; rrep(i, n) { (*this)[i] *= c; for (auto &[j, b] : g) { if (j > i) break; (*this)[i] += (*this)[i - j] * b; } } return *this; } F multiply(const vector> &g) const { return F(*this).multiply_inplace(g); } // O(nk) F ÷_inplace(vector> g) { int n = this->size(); auto[d, c] = g.front(); assert(d == 0 && c != T(0)); T ic = c.inv(); g.erase(g.begin()); rep(i, n) { for (auto &[j, b] : g) { if (j > i) break; (*this)[i] -= (*this)[i - j] * b; } (*this)[i] *= ic; } return *this; } F divide(const vector> &g) const { return F(*this).divide_inplace(g); } // multiply and divide (1 + cz^d) // O(n) void multiply_inplace(const int d, const T c) { int n = this->size(); if (c == T(1)) rrep(i, n - d) (*this)[i + d] += (*this)[i]; else if (c == T(-1)) rrep(i, n - d) (*this)[i + d] -= (*this)[i]; else rrep(i, n - d) (*this)[i + d] += (*this)[i] * c; } // O(n) void divide_inplace(const int d, const T c) { int n = this->size(); if (c == T(1)) rep(i, n - d) (*this)[i + d] -= (*this)[i]; else if (c == T(-1)) rep(i, n - d) (*this)[i + d] += (*this)[i]; else rep(i, n - d) (*this)[i + d] -= (*this)[i] * c; } // O(n) T eval(const T &a) const { T x(1), res(0); for (auto e : *this) res += e * x, x *= a; return res; } // O(n) F &integ_inplace() { int n = this->size(); assert(n > 0); if (n == 1) return *this = F{ 0 }; this->insert(this->begin(), 0); this->pop_back(); vector inv(n); inv[1] = 1; int p = T::mod(); rep2(i, 2, n) inv[i] = -inv[p%i] * (p / i); rep2(i, 2, n) (*this)[i] *= inv[i]; return *this; } F integ() const { return F(*this).integ_inplace(); } // O(n) F &deriv_inplace() { int n = this->size(); assert(n > 0); rep2(i, 2, n) (*this)[i] *= i; this->erase(this->begin()); this->push_back(0); return *this; } F deriv() const { return F(*this).deriv_inplace(); } // O(n log n) F &log_inplace(int d = -1) { int n = this->size(); assert(n > 0 && (*this)[0] == 1); if (d == -1) d = n; assert(d >= 0); if (d < n) this->resize(d); F f_inv = this->inv(); this->deriv_inplace(); this->multiply_inplace(f_inv); this->integ_inplace(); return *this; } F log(const int d = -1) const { return F(*this).log_inplace(d); } // O(n log n) // https://arxiv.org/abs/1301.5804 (Figure 1, right) F &exp_inplace(int d = -1) { int n = this->size(); assert(n > 0 && (*this)[0] == 0); if (d == -1) d = n; assert(d >= 0); F g{ 1 }, g_fft{ 1, 1 }; (*this)[0] = 1; this->resize(d); F h_drv(this->deriv()); for (int m = 2; m < d; m *= 2) { // prepare F f_fft(this->begin(), this->begin() + m); f_fft.resize(2 * m), internal::butterfly(f_fft); // Step 2.a' // { F _g(m); rep(i, m) _g[i] = f_fft[i] * g_fft[i]; internal::butterfly_inv(_g); _g.erase(_g.begin(), _g.begin() + m / 2); _g.resize(m), internal::butterfly(_g); rep(i, m) _g[i] *= g_fft[i]; internal::butterfly_inv(_g); _g.resize(m / 2); _g /= T(-m) * m; g.insert(g.end(), _g.begin(), _g.begin() + m / 2); // } // Step 2.b'--d' F t(this->begin(), this->begin() + m); t.deriv_inplace(); // { // Step 2.b' F r{ h_drv.begin(), h_drv.begin() + m - 1 }; // Step 2.c' r.resize(m); internal::butterfly(r); rep(i, m) r[i] *= f_fft[i]; internal::butterfly_inv(r); r /= -m; // Step 2.d' t += r; t.insert(t.begin(), t.back()); t.pop_back(); // } // Step 2.e' if (2 * m < d) { t.resize(2 * m); internal::butterfly(t); g_fft = g; g_fft.resize(2 * m); internal::butterfly(g_fft); rep(i, 2 * m) t[i] *= g_fft[i]; internal::butterfly_inv(t); t.resize(m); t /= 2 * m; } else { // この場合分けをしても数パーセントしか速くならない F g1(g.begin() + m / 2, g.end()); F s1(t.begin() + m / 2, t.end()); t.resize(m / 2); g1.resize(m), internal::butterfly(g1); t.resize(m), internal::butterfly(t); s1.resize(m), internal::butterfly(s1); rep(i, m) s1[i] = g_fft[i] * s1[i] + g1[i] * t[i]; rep(i, m) t[i] *= g_fft[i]; internal::butterfly_inv(t); internal::butterfly_inv(s1); rep(i, m / 2) t[i + m / 2] += s1[i]; t /= m; } // Step 2.f' F v(this->begin() + m, this->begin() + min(d, 2 * m)); v.resize(m); t.insert(t.begin(), m - 1, 0); t.push_back(0); t.integ_inplace(); rep(i, m) v[i] -= t[m + i]; // Step 2.g' v.resize(2 * m); internal::butterfly(v); rep(i, 2 * m) v[i] *= f_fft[i]; internal::butterfly_inv(v); v.resize(m); v /= 2 * m; // Step 2.h' rep(i, min(d - m, m)) (*this)[m + i] = v[i]; } return *this; } F exp(const int d = -1) const { return F(*this).exp_inplace(d); } // O(n log n) F &pow_inplace(const long long k, int d = -1) { int n = this->size(); if (d == -1) d = n; assert(d >= 0 && k >= 0); if (k == 0) { *this = F(d); if (d > 0) (*this)[0] = 1; return *this; } int l = 0; while (l < n && (*this)[l] == 0) ++l; if (l > (d - 1) / k || l == n) return *this = F(d); T c = (*this)[l]; this->erase(this->begin(), this->begin() + l); *this /= c; this->log_inplace(d - l * k); *this *= k; this->exp_inplace(); *this *= c.pow(k); this->insert(this->begin(), l*k, 0); return *this; } F pow(const long long k, const int d = -1) const { return F(*this).pow_inplace(k, d); } // O(n log n) F &shift_inplace(const T c) { int n = this->size(); fc = Factorial(n); rep(i, n) (*this)[i] *= fc.fac[i]; reverse(this->begin(), this->end()); F g(n); T cp = 1; rep(i, n) g[i] = cp * fc.finv[i], cp *= c; this->multiply_inplace(g, n); reverse(this->begin(), this->end()); rep(i, n) (*this)[i] *= fc.finv[i]; return *this; } F shift(const T c) const { return F(*this).shift_inplace(c); } F operator*(const T &g) const { return F(*this) *= g; } F operator/(const T &g) const { return F(*this) /= g; } F operator+(const F &g) const { return F(*this) += g; } F operator-(const F &g) const { return F(*this) -= g; } F operator<<(const int d) const { return F(*this) <<= d; } F operator>>(const int d) const { return F(*this) >>= d; } F operator*(vector> g) const { return F(*this) *= g; } F operator/(vector> g) const { return F(*this) /= g; } }; using fps = FormalPowerSeries; using sfps = vector>; // https://judge.yosupo.jp/problem/inv_of_formal_power_series // f(x)に対して 1/f(x)の先頭n項を求める void yosupo_inv() { int n; cin >> n; fps a(n); cin >> a; cout << a.inv() << '\n'; } // https://judge.yosupo.jp/problem/log_of_formal_power_series // f(x)に対して log(f(x))の先頭n項を求める void yosupo_log() { int n; cin >> n; fps a(n); cin >> a; cout << a.log_inplace() << '\n'; } // https://judge.yosupo.jp/problem/exp_of_formal_power_series // f(x)に対して exp(f(x))の先頭n項を求める void yosupo_exp() { int n; cin >> n; fps a(n); cin >> a; cout << a.exp_inplace() << '\n'; } // https://judge.yosupo.jp/problem/pow_of_formal_power_series // f(x)に対してf(x)^mの先頭n項を求める void yosupo_pow() { int n, m; cin >> n >> m; fps a(n); cin >> a; cout << a.pow_inplace(m) << '\n'; } // https://judge.yosupo.jp/problem/polynomial_taylor_shift // f(x)に対してf(x+c)の先頭n項を求める void yosupo_shift() { int n, c; cin >> n >> c; fps a(n); cin >> a; cout << a.shift_inplace(c) << '\n'; } // https://judge.yosupo.jp/problem/bernoulli_number // ベルヌーイ数の先頭n項を求める void yosupo_bernoulli() { int n; cin >> n; ++n; fc = Factorial(n); fps f((fc.finv).begin() + 1, (fc.finv).end()); f = f.inv(); rep(i, n) f[i] *= fc.fac[i]; cout << f << '\n'; } // https://judge.yosupo.jp/problem/sharp_p_subset_sum // Π(1+x^a)の先頭t+1項を求める void yosupo_count_subset_sum() { int n, t; cin >> n >> t; ++t; vector c(t); rep(i, n) { int s; cin >> s; ++c[s]; } vector inv(t); inv[1] = 1; int p = mint::mod(); rep2(i, 2, t) inv[i] = -inv[p%i] * (p / i); fps f(t); rep(i, t) { if (c[i] == 0) continue; for (int j = 1, d = i; d < t; ++j, d += i) { if (j & 1) f[d] += c[i] * inv[j]; else f[d] -= c[i] * inv[j]; } } f.exp_inplace(); f.erase(f.begin()); cout << f << '\n'; } int main() { ios::sync_with_stdio(false); cin.tie(nullptr); int n; cin >> n; vectorp(n); rep(i, n)cin >> p[i]; vectorf(n + 1), g(n + 1); f[0].resize(n + 1); f[0][0] = 1; g[0].resize(n + 1); g[0][0] = 1; rep(i, n) { sfps h = { {0,1},{1,p[i]} }; f[i + 1] = f[i]; f[i + 1].multiply_inplace(h); g[i + 1] = g[i]; g[i + 1].divide_inplace(h); } int q; cin >> q; while (q--) { int a, b, k; cin >> a >> b >> k; a--, b--; mint ans = 0; rep(i, k + 1) { ans += f[b + 1][i] * g[a][k - i]; } cout << ans.val() << endl; } return 0; }