// May this submission get accepted #include // エイリアス using ll = long signed long; using ull = long unsigned long; using ld = long double; using namespace std; // エイリアス (補完・コンパイルが重くなる) // #include // using mll = boost::multiprecision::cpp_int; // 汎用マクロ #define ALL_OF(x) (x).begin(), (x).end() #define REP(i,n) for (long long i=0, i##_len=(n); i=i##_end; i--) #define STEP(i, is, ie, step) for (long long i=(is), i##_end=(ie), i##_step = (step); i<=i##_end; i+=i##_step) #define UNIQUE(v) do { sort((v).begin(), (v).end()); (v).erase(unique((v).begin(), (v).end()), (v).end()); } while (false) #define FOREACH(i,q) for (auto &i : q) template bool chmax(T &a, const T b) { if (a < b) {a = b; return true;} return false; } template bool chmin(T &a, const T b) { if (a > b) {a = b; return true;} return false; } constexpr int INF = numeric_limits::max(); constexpr long long LINF = numeric_limits::max(); #define Yes(q) ((q) ? "Yes" : "No") #define YES(q) ((q) ? "YES" : "NO") #define Possible(q) ((q) ? "Possible" : "Impossible") #define POSSIBLE(q) ((q) ? "POSSIBLE" : "IMPOSSIBLE") #define DUMP(q) DUMP_FUNC(q, #q, __FILE__, __LINE__) template void DUMP_PROC(T x) { cerr << x; } void DUMP_PROC(string x) { cerr << '"' << x << '"'; } template void DUMP_PROC(vector x) { cerr << "["; for (auto &xi : x) { DUMP_PROC(xi); cerr << (&xi != &*x.rbegin()?", ":""); } cerr << "]"; } template void DUMP_FUNC(T x, const char* name, const char* fn, int ln) { cerr << "\e[32m[DEBUG]\e[0m " << name << ": "; DUMP_PROC(x); cerr << " @ " << fn << "(" << ln << ")" << endl; } template T gcd(T a, T b) { if (a < b) swap(a, b); while (b) swap(a %= b, b); return a; } template T lcm(const T a, const T b) { return a / gcd(a, b) * b; } // gcc拡張マクロ #define popcount __builtin_popcount #define popcountll __builtin_popcountll // 標準入出力 struct inp { size_t sz; inp(size_t _sz = 1) : sz(_sz) {} template operator T () const { T a; cin >> a; return a; } template operator vector () const { vector a(sz); for (size_t i = 0; i < sz; i++) cin >> a[i]; return a; } template operator pair () const { T f; U s; cin >> f >> s; return pair(f, s); } }; inp inp1; // input one template void say(const T x, const char* end = "\n") { cout << x << end; } void say(const ld x, const char* end = "\n") { cout << setprecision(30) << x << end; } template void say(const vector x, const char* sep = " ", const char* end = "\n") { REP(i, x.size()) { cout << x[i] << (i+1 == i_len ? end : sep); } } template void say(const vector> x, const char* sep = " ", const char* end = "\n") { REP(i, x.size()) { say(x[i], sep, end); } } // モジュール // 精度に寄りけりだがconv1回で済むFFTの方がいい場合もあることに留意 // Cooley-Tukey型 高速フーリエ変換 O(NlogN) template vector ntt(vector a, const bool inv = false) { const auto modpow = [](ll a, ll n) -> ll { ll r = 1; ((n %= pdiv-1) += pdiv-1) %= pdiv-1; for (ll b = (a % pdiv + pdiv) % pdiv; n; n >>= 1, (b *= b) %= pdiv) { if (n & 1) (r *= b) %= pdiv; } return r; }; // Shita-goshirae ll n = 1, h = 0; while (n < a.size()) n <<= 1, h++; a.resize(n, 0); for (size_t i = 0; i < n; i++) { ((a[i] %= pdiv) += pdiv) %= pdiv; } // Cooley-Tukey sort for (size_t i = 0; i < n; i++) { size_t j = 0; for (size_t k = 0; k < h; k++) { j |= (i >> k & 1) << (h-1 - k); } if (i < j) swap(a[i], a[j]); } // butterfly diagram for (size_t b = 1; b < n; b <<= 1) { const ll root = modpow(prim, (pdiv-1) / (b << 1) * (inv ? -1 : 1)); ll w = 1; for (size_t j = 0; j < b; j++) { for (size_t k = 0; k < n; k += b << 1) { ll s = a[j+k ]; ll t = a[j+k+b] * w % pdiv; a[j+k ] = (s+t) % pdiv; a[j+k+b] = (s-t + pdiv) % pdiv; } (w *= root) %= pdiv; } } if (inv) { ll invn = modpow(n, pdiv-2); for (ll i = n; i--; ) (a[i] *= invn) %= pdiv; } return a; } // 適切な原始根の存在する法での畳み込み template vector convolve_p(const vector &x, const vector &y) { size_t t = x.size() + y.size() - 1; auto a = x; a.resize(t, 0); auto b = y; b.resize(t, 0); auto A = ntt(a); auto B = ntt(b); for (size_t i = 0; i < A.size(); i++) (A[i] *= B[i]) %= pdiv; auto result = ntt(A, true); result.resize(t); return result; } // 任意MODでの畳み込み演算 vector convolve(const vector &x, const vector &y, const ll pdiv) { if (pdiv == 998244353) return convolve_p<998244353, 3>(x, y); // 1. 配列の大きさが 2^23 ~= 8e6 超えたら1の2^N乗根求まらなくてバグる // 競プロならO(N)すらTLEしかねないから大丈夫 // conv2をコメントアウトするとこの上限が2^24 ~= 1.6e7ほどに伸びる // 2. 答えの大きさが Πm ~= 9.6e34 超えたらGarnerがバグる // すなわち配列x, yの各項が √(Πm/2^23) ~= 1e14 超えてるとバグる // 競プロなら大抵法が1e9くらいなので大丈夫 // conv2をコメントアウトすると上限が 2.398e9 ほどになるが1e9+7とかならセーフ // 3. 出典: https://lumakernel.github.io/ecasdqina/math/FFT/NTT // {1224736769, 3}, // 2^24 * 73 + 1 // {998244353, 3}, // 2^23 * 7 * 17 + 1 // {469762049, 3}, // 2^26 * 7 + 1 // {167772161, 3}, // 2^25 * 5 + 1 const auto conv1 = convolve_p<1224736769, 3>(x, y); // const auto conv2 = convolve_p<998244353, 3>(x, y); const auto conv3 = convolve_p<469762049, 3>(x, y); const auto conv4 = convolve_p<167772161, 3>(x, y); const auto garner = [&](const vector> &w) -> ll { const auto modpow = [](ll a, ll n, ll pdiv) -> ll { ll r = 1; ((n %= pdiv-1) += pdiv-1) %= pdiv-1; for (ll b = (a % pdiv + pdiv) % pdiv; n; n >>= 1, (b *= b) %= pdiv) { if (n & 1) (r *= b) %= pdiv; } return r; }; vector p(w.size()+1, 1); vector v(w.size()+1, 0); for (size_t k = 0; k < w.size(); k++) { ll t = (w[k].first - v[k]) * modpow(p[k], w[k].second-2, w[k].second); ((t %= w[k].second) += w[k].second) %= w[k].second; for (size_t i = k+1; i <= w.size(); i++) { ll bi = i == w.size() ? pdiv : w[i].second; (v[i] += t * p[i]) %= bi; (p[i] *= w[k].second) %= bi; } } return v.back(); }; ll n = conv1.size(); ll t = x.size() + y.size() - 1; vector result(t, 0); REP(i, t) { vector> g = { {conv1[i], 1224736769}, // {conv2[i], 998244353}, {conv3[i], 469762049}, {conv4[i], 167772161} }; result[i] = garner(g); } return result; } // 形式的冪級数の逆元のdeg項まで(deg-1次の項まで)を返す vector formalinv(const vector &x, ll deg, ll pdiv) { assert(x.front() != 0); const auto modpow = [&](ll a, ll n) -> ll { ll r = 1; ((n %= pdiv-1) += pdiv-1) %= pdiv-1; for (ll b = (a % pdiv + pdiv) % pdiv; n; n >>= 1, (b *= b) %= pdiv) { if (n & 1) (r *= b) %= pdiv; } return r; }; const auto modinv = [&](ll a) { return modpow(a, pdiv-2); }; vector rs = {modinv(x[0])}; for (size_t i = 1; i < deg; i *= 2) { auto rs2 = rs; for (ll &rsi : rs2) (rsi *= 2) %= pdiv; auto rss = convolve(rs, rs, pdiv); vector as(i*2, 0); REP(j, min(i*2, x.size())) as[j] = (x[j] % pdiv + pdiv) % pdiv; auto rcv = convolve(rss, as, pdiv); rs .resize(i*2, 0); rs2.resize(i*2, 0); rcv.resize(i*2, 0); for (size_t j = 0; j < i*2; j++) { rs[j] = (rs2[j] - rcv[j] + pdiv) % pdiv; } } return rs; } // 処理内容 int main() { ios::sync_with_stdio(false); // stdioを使うときはコメントアウトすること cin.tie(nullptr); // インタラクティブ問題ではコメントアウトすること constexpr ll pdiv = 1000000007; ll n = inp1; ll w = inp1; ll k = inp1; vector a = inp(n); vector dp(w*2+1, 0); dp[0]++; for (ll ai : a) dp[ai]--; dp = formalinv(dp, w*2+1, pdiv); dp.resize(w*2+1); ll dp1 = dp[w]; ll dp2 = (dp[w*2] - dp1*dp1%pdiv + pdiv) % pdiv; ll m[2][2] = {}; auto mtrprod = [&](vector> a, vector> b) { vector> r = {{0, 0}, {0, 0}}; REP(i, 2) REP(j, 2) REP(k, 2) (r[i][j] += a[i][k] * b[k][j] % pdiv) %= pdiv; return r; }; auto mtrpow = [&](vector> a, ll n) -> vector> { vector> r = {{1, 0}, {0, 1}}; for (vector> b = a; n; n >>= 1) { if (n & 1) { r = mtrprod(r, b); } b = mtrprod(b, b); } return r; }; vector> d = { {dp1, dp2}, { 1, 0} }; d = mtrpow(d, k); say(d[0][0]); }