#include using namespace std; #include using namespace atcoder; // input and output of modint 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 ostream &operator<<(ostream &os, const static_modint &a) { return os << a.val(); } template istream &operator>>(istream &is, dynamic_modint &a) { long long v; is >> v; a = v; return is; } template ostream &operator<<(ostream &os, const dynamic_modint &a) { return os << a.val(); } #define rep_(i, a_, b_, a, b, ...) for (int i = (a), lim##i = (b); i < lim##i; ++i) #define rep(i, ...) rep_(i, __VA_ARGS__, __VA_ARGS__, 0, __VA_ARGS__) #define drep_(i, a_, b_, a, b, ...) for (int i = (a)-1, lim##i = (b); i >= lim##i; --i) #define drep(i, ...) drep_(i, __VA_ARGS__, __VA_ARGS__, __VA_ARGS__, 0) #define all(x) (x).begin(), (x).end() #define rall(x) (x).rbegin(), (x).rend() #ifdef LOCAL void debug_out() { cerr << endl; } template void debug_out(Head H, Tail... T) { cerr << ' ' << H; debug_out(T...); } #define debug(...) cerr << 'L' << __LINE__ << " [" << #__VA_ARGS__ << "]:", debug_out(__VA_ARGS__) #define dump(x) cerr << 'L' << __LINE__ << " " << #x << " = " << (x) << endl #else #define debug(...) (void(0)) #define dump(x) (void(0)) #endif template using V = vector; using ll = long long; using ld = long double; using Vi = V; using VVi = V; using Vl = V; using VVl = V; using Vd = V; using VVd = V; using Vb = V; using VVb = V; template using priority_queue_rev = priority_queue, greater>; template vector make_vec(size_t n, T a) { return vector(n, a); } template auto make_vec(size_t n, Ts... ts) { return vector(n, make_vec(ts...)); } template inline int sz(const T &x) { return x.size(); } template inline bool chmin(T &a, const T b) { if (a > b) { a = b; return true; } return false; } template inline bool chmax(T &a, const T b) { if (a < b) { a = b; return true; } return false; } template istream &operator>>(istream &is, pair &p) { is >> p.first >> p.second; return is; } template ostream &operator<<(ostream &os, const pair &p) { os << '(' << p.first << ", " << p.second << ')'; return os; } template istream &operator>>(istream &is, array &v) { for (auto &e : v) is >> e; return is; } template ostream &operator<<(ostream &os, const array &v) { for (auto &e : v) os << e << ' '; return os; } 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 inline void deduplicate(vector &a) { sort(all(a)); a.erase(unique(all(a)), a.end()); } template inline int count_between(const vector &a, T l, T r) { return lower_bound(all(a), r) - lower_bound(all(a), l); } // [l, r) inline ll ceil_div(const ll x, const ll y) { return (x+y-1) / y; } // ceil(x/y) inline int floor_log2(const ll x) { assert(x > 0); return 63-__builtin_clzll(x); } // floor(log2(x)) inline int ceil_log2(const ll x) { assert(x > 0); return (x == 1) ? 0 : 64-__builtin_clzll(x-1); } // ceil(log2(x)) inline int popcount(const ll x) { return __builtin_popcountll(x); } inline void fail() { cout << -1 << '\n'; exit(0); } struct fast_ios { fast_ios(){ cin.tie(nullptr); ios::sync_with_stdio(false); cout << fixed << setprecision(20); }; } fast_ios_; // const int INF = (1<<30) - 1; // const ll INFll = (1ll<<60) - 1; // const ld EPS = 1e-10; // const ld PI = acos(-1.0); // using mint = modint998244353; using mint = modint1000000007; // using mint = modint; using Vm = V; using VVm = V; template struct Factorial { int MAX; vector fac, finv; Factorial(int m = 0) : MAX(m), fac(m+1, 1), finv(m+1, 1) { rep(i, 2, MAX+1) fac[i] = fac[i-1] * i; finv[MAX] /= fac[MAX]; drep(i, MAX+1, 3) 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; template struct formal_power_series : vector { using vector::vector; using vector::operator=; using F = formal_power_series; 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(); // drep(i, n) { // (*this)[i] *= g[0]; // rep(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) { // rep(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; drep(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)) drep(i, n-d) (*this)[i+d] += (*this)[i]; else if (c == T(-1)) drep(i, n-d) (*this)[i+d] -= (*this)[i]; else drep(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 &integral_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(); rep(i, 2, n) inv[i] = - inv[p%i] * (p/i); rep(i, 2, n) (*this)[i] *= inv[i]; return *this; } F integral() const { return F(*this).integral_inplace(); } // O(n) F &derivative_inplace() { int n = this->size(); assert(n > 0); rep(i, 2, n) (*this)[i] *= i; this->erase(this->begin()); this->push_back(0); return *this; } F derivative() const { return F(*this).derivative_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->derivative_inplace(); this->multiply_inplace(f_inv); this->integral_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->derivative()); 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.derivative_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.integral_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 ll k, int d = -1) { int n = this->size(); if (d == -1) d = n; assert(d >= 0 && k >= 0); if (d == 0) return *this = F(0); if (k == 0) { *this = F(d); (*this)[0] = 1; return *this; } int l = 0; while (l < n && (*this)[l] == 0) ++l; if (l == n || l > (d-1)/k) 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 ll 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; } // for multivariable FPS // F operator*(const F &g) const { return this->multiply(g); } // F operator*=(const F &g) { return this->multiply_inplace(g); } }; using fps = formal_power_series; mint solve() { int n, mp, mq, l; cin >> n >> mp >> mq >> l; fc = Factorial(n + mp + mq); V f(n+1, fps(mq+1)); f[0][0] = 1; rep(i, n) { int s; cin >> s; drep(j, n) { auto g = f[j]; g.multiply_inplace(s, -1); g.divide_inplace(1, -1); g <<= 1; f[j+1] += g; } } mint ans = 0; rep(i, n+1) rep(j, mq+1) { int dz = mq - j; int dx = mp + mq - dz - l*i; ans += f[i][j] * fc.binom(n + dx - 1, dx); } return ans; } int main() { // solve(); cout << solve() << '\n'; }