#pragma region Macros // #pragma GCC target("avx2") #pragma GCC optimize("O3") // #pragma GCC optimize("unroll-loops") #include #define ll long long #define ld long double #define rep2(i, a, b) for(ll i = a; i <= b; ++i) #define rep(i, n) for(ll i = 0; i < n; ++i) #define rep3(i, a, b) for(ll i = a; i >= b; --i) #define pii pair #define pll pair #define pb push_back #define eb emplace_back #define vi vector #define vll vector #define vpi vector #define vpll vector #define overload2(_1, _2, name, ...) name #define vec(type, name, ...) vector name(__VA_ARGS__) #define VEC(type, name, size) \ vector name(size); \ IN(name) #define vv(type, name, h, ...) vector> name(h, vector(__VA_ARGS__)) #define VV(type, name, h, w) \ vector> name(h, vector(w)); \ IN(name) #define vvv(type, name, h, w, ...) vector>> name(h, vector>(w, vector(__VA_ARGS__))) #define vvvv(type, name, a, b, c, ...) \ vector>>> name(a, vector>>(b, vector>(c, vector(__VA_ARGS__)))) #define mt make_tuple #define fi first #define se second #define all(c) begin(c), end(c) #define SUM(v) accumulate(all(v), 0LL) #define MIN(v) *min_element(all(v)) #define MAX(v) *max_element(all(v)) #define lb(c, x) distance((c).begin(), lower_bound(all(c), (x))) #define ub(c, x) distance((c).begin(), upper_bound(all(c), (x))) using namespace std; constexpr pii dx4[4] = {pii{1, 0}, pii{0, 1}, pii{-1, 0}, pii{0, -1}}; constexpr pii dx8[8] = {{1, 0}, {1, 1}, {0, 1}, {-1, 1}, {-1, 0}, {-1, -1}, {0, -1}, {1, -1}}; const string YESNO[2] = {"NO", "YES"}; const string YesNo[2] = {"No", "Yes"}; const string yesno[2] = {"no", "yes"}; void YES(bool t = 1) { cout << YESNO[t] << endl; } void Yes(bool t = 1) { cout << YesNo[t] << endl; } void yes(bool t = 1) { cout << yesno[t] << endl; } template using vc = vector; template using vvc = vector>; template using vvvc = vector>; template using vvvvc = vector>; template using pq = priority_queue; template using pqg = priority_queue, greater>; #define si(c) (int)(c).size() #define INT(...) \ int __VA_ARGS__; \ IN(__VA_ARGS__) #define LL(...) \ ll __VA_ARGS__; \ IN(__VA_ARGS__) #define STR(...) \ string __VA_ARGS__; \ IN(__VA_ARGS__) #define CHR(...) \ char __VA_ARGS__; \ IN(__VA_ARGS__) #define DBL(...) \ double __VA_ARGS__; \ IN(__VA_ARGS__) int scan() { return getchar(); } void scan(int &a) { cin >> a; } void scan(long long &a) { cin >> a; } void scan(char &a) { cin >> a; } void scan(double &a) { cin >> a; } void scan(string &a) { cin >> a; } template void scan(pair &p) { scan(p.first), scan(p.second); } template void scan(vector &); template void scan(vector &a) { for(auto &i : a) scan(i); } template void scan(T &a) { cin >> a; } void IN() {} template void IN(Head &head, Tail &...tail) { scan(head); IN(tail...); } template inline bool chmax(T &a, const S &b) { return (a < b ? a = b, 1 : 0); } template inline bool chmin(T &a, const S &b) { return (a > b ? a = b, 1 : 0); } vi iota(int n) { vi a(n); iota(all(a), 0); return a; } template vi iota(vector &a, bool greater = false) { vi res(a.size()); iota(all(res), 0); sort(all(res), [&](int i, int j) { if(greater) return a[i] > a[j]; return a[i] < a[j]; }); return res; } #define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()) template T POW(T x, int n) { T res = 1; for(; n; n >>= 1, x *= x) if(n & 1) res *= x; return res; } vector factor(ll x) { vector ans; for(ll i = 2; i * i <= x; i++) if(x % i == 0) { ans.push_back({i, 1}); while((x /= i) % i == 0) ans.back().second++; } if(x != 1) ans.push_back({x, 1}); return ans; } template vector divisor(T x) { vector ans; for(T i = 1; i * i <= x; i++) if(x % i == 0) { ans.pb(i); if(i * i != x) ans.pb(x / i); } return ans; } template void zip(vector &x) { vector y = x; sort(all(y)); for(int i = 0; i < x.size(); ++i) { x[i] = lb(y, x[i]); } } int topbit(signed t) { return t == 0 ? -1 : 31 - __builtin_clz(t); } int topbit(ll t) { return t == 0 ? -1 : 63 - __builtin_clzll(t); } int lowbit(signed a) { return a == 0 ? 32 : __builtin_ctz(a); } int lowbit(ll a) { return a == 0 ? 64 : __builtin_ctzll(a); } // int allbit(int n) { return (1 << n) - 1; } ll allbit(ll n) { return (1LL << n) - 1; } int popcount(signed t) { return __builtin_popcount(t); } int popcount(ll t) { return __builtin_popcountll(t); } bool ispow2(int i) { return i && (i & -i) == i; } int in() { int x; cin >> x; return x; } ll lin() { unsigned long long x; cin >> x; return x; } template pair operator-(const pair &x, const pair &y) { return pair(x.fi - y.fi, x.se - y.se); } template pair operator+(const pair &x, const pair &y) { return pair(x.fi + y.fi, x.se + y.se); } // template pair &operator+=(pair x, const pair &y) { // x = x + y; // return &x; // } // template pair &operator-=(pair x, const pair &y) { // x = x - y; // return &x; // } template ll operator*(const pair &x, const pair &y) { return (ll)x.fi * y.fi + (ll)x.se * y.se; } template struct edge { int from, to; T cost; int id; edge(int to, T cost) : from(-1), to(to), cost(cost) {} edge(int from, int to, T cost) : from(from), to(to), cost(cost) {} edge(int from, int to, T cost, int id) : from(from), to(to), cost(cost), id(id) {} edge &operator=(const int &x) { to = x; return *this; } operator int() const { return to; } }; template using Edges = vector>; using Tree = vector>; using Graph = vector>; template using Wgraph = vector>>; Graph getG(int n, int m = -1, bool directed = false, int margin = 1) { Tree res(n); if(m == -1) m = n - 1; while(m--) { int a, b; cin >> a >> b; a -= margin, b -= margin; res[a].emplace_back(b); if(!directed) res[b].emplace_back(a); } return move(res); } template Wgraph getWg(int n, int m = -1, bool directed = false, int margin = 1) { Wgraph res(n); if(m == -1) m = n - 1; while(m--) { int a, b; T c; cin >> a >> b >> c; a -= margin, b -= margin; res[a].emplace_back(b, c); if(!directed) res[b].emplace_back(a, c); } return move(res); } #define i128 __int128_t #define ull unsigned long long int #define TEST \ INT(testcases); \ while(testcases--) template ostream &operator<<(ostream &os, const vector &v) { for(auto it = begin(v); it != end(v); ++it) { if(it == begin(v)) os << *it; else os << " " << *it; } return os; } template ostream &operator<<(ostream &os, const pair &p) { os << p.first << " " << p.second; return os; } template string to_string(pair p) { return "(" + to_string(p.first) + "," + to_string(p.second) + ")"; } template string to_string(A v) { if(v.empty()) return "{}"; string ret = "{"; for(auto &x : v) ret += to_string(x) + ","; ret.back() = '}'; return ret; } string to_string(string s) { return "\"" + s + "\""; } void dump() { cerr << endl; } template void dump(Head head, Tail... tail) { cerr << to_string(head) << " "; dump(tail...); } #define endl '\n' #ifdef _LOCAL #undef endl #define debug(x) \ cout << #x << ": "; \ dump(x) #else #define debug(x) #endif template static constexpr T inf = numeric_limits::max() / 2; struct Setup_io { Setup_io() { ios_base::sync_with_stdio(0), cin.tie(0), cout.tie(0); cout << fixed << setprecision(15); } } setup_io; #define drop(s) cout << #s << endl, exit(0) #pragma endregion namespace modular { constexpr ll MOD = 998244353; const int MAXN = 1100000; template class modint; #define mint modint #define vmint vector vector Inv; mint inv(int x); template class modint { public: static constexpr int mod() { return Modulus; } ll a; constexpr modint(const ll x = 0) noexcept : a(((x % Modulus) + Modulus) % Modulus) {} constexpr ll &value() noexcept { return a; } constexpr const ll &value() const noexcept { return a; } constexpr modint operator-() const noexcept { return modint() - *this; } constexpr modint operator+() const noexcept { return *this; } constexpr modint &operator++() noexcept { if(++a == MOD) a = 0; return *this; } constexpr modint &operator--() noexcept { if(!a) a = MOD; a--; return *this; } constexpr modint operator++(int) { modint res = *this; ++*this; return res; } constexpr modint operator--(int) { mint res = *this; --*this; return res; } constexpr modint &operator+=(const modint rhs) noexcept { a += rhs.a; if(a >= Modulus) { a -= Modulus; } return *this; } constexpr modint &operator-=(const modint rhs) noexcept { if(a < rhs.a) { a += Modulus; } a -= rhs.a; return *this; } constexpr modint &operator*=(const modint rhs) noexcept { a = a * rhs.a % Modulus; return *this; } constexpr modint &operator/=(const modint rhs) noexcept { a = a * (modular::inv(rhs.a)).a % Modulus; return *this; } constexpr modint pow(long long n) const noexcept { if(n < 0) { n %= Modulus - 1; n = (Modulus - 1) + n; } modint x = *this, r = 1; while(n) { if(n & 1) r *= x; x *= x; n >>= 1; } return r; } constexpr modint inv() const noexcept { return pow(Modulus - 2); } constexpr friend modint operator+(const modint &lhs, const modint &rhs) { return modint(lhs) += modint(rhs); } constexpr friend modint operator-(const modint &lhs, const modint &rhs) { return modint(lhs) -= modint(rhs); } constexpr friend modint operator*(const modint &lhs, const modint &rhs) { return modint(lhs) *= modint(rhs); } constexpr friend modint operator/(const modint &lhs, const modint &rhs) { return modint(lhs) /= modint(rhs); } constexpr friend modint operator==(const modint &lhs, const modint &rhs) { return lhs.a == rhs.a; } constexpr friend modint operator!=(const modint &lhs, const modint &rhs) { return lhs.a != rhs.a; } // constexpr friend modint operator^=(const modint &lhs, const modint &rhs) { return modint(lhs) ^= modint(rhs); } }; vmint Fact{1, 1}, Ifact{1, 1}; mint inv(int n) { if(n > MAXN) return (mint(n)).pow(MOD - 2); if(Inv.empty()) Inv.emplace_back(0), Inv.emplace_back(1); if(Inv.size() > n) return Inv[n]; else { Inv.reserve(n + 1); for(int i = Inv.size(); i <= n; ++i) { auto [y, x] = div(int(MOD), i); Inv.emplace_back(Inv[x] * (-y)); } return Inv[n]; } } mint fact(int n) { if(Fact.size() > n) return Fact[n]; else for(int i = Fact.size(); i <= n; ++i) Fact.emplace_back(Fact[i - 1] * i); return Fact[n]; } mint ifact(int n) { if(Ifact.size() > n) return Ifact[n]; else for(int i = Ifact.size(); i <= n; ++i) Ifact.emplace_back(Ifact[i - 1] * inv(i)); return Ifact[n]; } mint modpow(ll a, ll n) { return mint(a).pow(n); } mint inv(mint a) { return inv(a.a); } mint ifact(mint a) { return ifact(a.a); } mint fact(mint a) { return fact(a.a); } mint modpow(mint a, ll n) { return modpow(a.a, n); } mint C(int a, int b) { if(a < 0 || b < 0) return 0; if(a < b) return 0; if(a > MAXN) { mint res = 1; rep(i, b) res *= a - i, res /= i + 1; return res; } return fact(a) * ifact(b) * ifact(a - b); } mint P(int a, int b) { if(a < 0 || b < 0) return 0; if(a < b) return 0; if(a > MAXN) { mint res = 1; rep(i, b) res *= a - i; return res; } return fact(a) * ifact(a - b); } ostream &operator<<(ostream &os, mint a) { os << a.a; return os; } istream &operator>>(istream &is, mint &a) { ll x; is >> x; a = x; return is; } mint proot = 3; void FMT(vmint &f, const bool is_inv = false) { const int n = f.size(); const mint root = is_inv ? inv(proot) : proot; vmint g(n); for(int b = n >> 1; b > 0; b >>= 1) { mint a = root.pow((MOD - 1) / (n / b)), p = 1; for(int i = 0; i < n; i += b << 1) { rep(j, b) { f[i + j + b] *= p; g[(i >> 1) + j] = f[i + j] + f[i + b + j]; g[(n >> 1) + (i >> 1) + j] = f[i + j] - f[i + b + j]; } p *= a; } swap(f, g); } if(is_inv) rep(i, n) f[i] *= inv(n); } vmint mul(vmint x, const vmint &y) { int n = x.size() + y.size() - 1; int s = 1; while(s < n) s <<= 1; x.resize(s); FMT(x); vmint z(s); rep(i, y.size()) z[i] = y[i]; FMT(z); rep(i, s) x[i] *= z[i]; FMT(x, true); x.resize(n); return x; } using Poly = vmint; Poly operator-(Poly f) { for(auto &&e : f) e = -e; return f; } Poly &operator+=(Poly &l, const Poly &r) { l.resize(max(l.size(), r.size())); rep(i, r.size()) l[i] += r[i]; return l; } Poly operator+(Poly l, const Poly &r) { return l += r; } Poly &operator-=(Poly &l, const Poly &r) { l.resize(max(l.size(), r.size())); rep(i, r.size()) l[i] -= r[i]; return l; } Poly operator-(Poly l, const Poly &r) { return l -= r; } Poly &operator<<=(Poly &f, size_t n) { return f.insert(f.begin(), n, 0), f; } Poly operator<<(Poly f, size_t n) { return f <<= n; } Poly &operator>>=(Poly &f, size_t n) { return f.erase(f.begin(), f.begin() + min(f.size(), n)), f; } Poly operator>>(Poly f, size_t n) { return f >>= n; } Poly operator*(const Poly &l, const Poly &r) { return mul(l, r); } Poly &operator*=(Poly &l, const Poly &r) { return l = l * r; } Poly inv(const Poly &f) { Poly g{1 / f[0]}; while(g.size() < f.size()) { Poly x(f.begin(), f.begin() + min(f.size(), g.size() << 1)), y = g; x.resize(g.size() << 1), FMT(x); y.resize(g.size() << 1), FMT(y); rep(i, x.size()) x[i] *= y[i]; FMT(x, true); x >>= g.size(); x.resize(g.size() << 1), FMT(x); rep(i, x.size()) x[i] *= -y[i]; FMT(x, true); g.insert(g.end(), x.begin(), x.begin() + g.size()); } return Poly{begin(g), begin(g) + f.size()}; } Poly integ(const Poly &f) { Poly res(f.size() + 1); for(int i = 1; i < (int)res.size(); ++i) res[i] = f[i - 1] / i; return res; } Poly deriv(const Poly &f) { if(f.size() == 0) return Poly(); Poly res(f.size() - 1); rep(i, res.size()) res[i] = f[i + 1] * (i + 1); return res; } Poly log(const Poly &f) { Poly g = integ(inv(f) * deriv(f)); return Poly{g.begin(), g.begin() + f.size()}; } Poly exp(const Poly &f) { Poly g{1}; while(g.size() < f.size()) { Poly x(f.begin(), f.begin() + min(f.size(), g.size() * 2)); x[0] += 1; g.resize(2 * g.size()); x -= log(g); x *= {g.begin(), g.begin() + g.size() / 2}; rep2(i, g.size() / 2, min(x.size(), g.size()) - 1) g[i] = x[i]; } return {g.begin(), g.begin() + f.size()}; } } // namespace modular using namespace modular; int main() { INT(n, k); mint factor = 1; VEC(int, M, n); vmint a; rep(i, n) { factor *= M[i]; a.eb((M[i] + 1) * inv(2)); } mint ans; { mint sum = accumulate(all(a), mint(0)); sum *= sum; for(auto e : a) sum -= e * e; sum /= 2; rep2(i, 1, k) { ans += P(n - 1, i) * sum; } } // cout << ans * factor << endl; { auto b = a; reverse(all(b)); auto f = a * b; mint t = 0; rep3(i, n * 2 - 3, n) { int d = n * 2 - 2 - i; t += ifact(d - 1); if(k + 1 <= d) t -= ifact(d - k - 1); ans -= f[i] * t * fact(d); } } cout << ans * factor << endl; }