#include using namespace std; #define LOCAL #pragma region Macros typedef long long ll; #define ALL(x) (x).begin(),(x).end() const long long MOD=1000000007; // const long long MOD=998244353; const int INF=1e9; const long long IINF=1e18; const int dx[4]={1,0,-1,0},dy[4]={0,1,0,-1}; const char dir[4]={'D','R','U','L'}; template istream &operator>>(istream &is,vector &v){ for (T &x:v) is >> x; return is; } template ostream &operator<<(ostream &os,const vector &v){ for (int i=0;i ostream &operator<<(ostream &os,const pair &p){ os << '(' << p.first << ',' << p.second << ')'; return os; } template ostream &operator<<(ostream &os,const map &m){ os << '{'; for (auto itr=m.begin();itr!=m.end();++itr){ os << '(' << itr->first << ',' << itr->second << ')'; if (++itr!=m.end()) os << ','; --itr; } os << '}'; return os; } template ostream &operator<<(ostream &os,const set &s){ os << '{'; for (auto itr=s.begin();itr!=s.end();++itr){ os << *itr; if (++itr!=s.end()) os << ','; --itr; } os << '}'; return os; } void debug_out(){cerr << '\n';} template void debug_out(Head&& head,Tail&&... tail){ cerr << head; if (sizeof...(Tail)>0) cerr << ", "; debug_out(move(tail)...); } #ifdef LOCAL #define debug(...) cerr << " ";\ cerr << #__VA_ARGS__ << " :[" << __LINE__ << ":" << __FUNCTION__ << "]" << '\n';\ cerr << " ";\ debug_out(__VA_ARGS__) #else #define debug(...) 42 #endif template T gcd(T x,T y){return y!=0?gcd(y,x%y):x;} template T lcm(T x,T y){return x/gcd(x,y)*y;} template inline bool chmin(T1 &a,T2 b){ if (a>b){a=b; return true;} return false; } template inline bool chmax(T1 &a,T2 b){ if (a class modint{ using i64=int64_t; using u32=uint32_t; using u64=uint64_t; public: u32 v; constexpr modint(const i64 x=0) noexcept :v(x<0?mod-1-(-(x+1)%mod):x%mod){} constexpr u32 &value() noexcept {return v;} constexpr const u32 &value() const noexcept {return v;} constexpr modint operator+(const modint &rhs) const noexcept {return modint(*this)+=rhs;} constexpr modint operator-(const modint &rhs) const noexcept {return modint(*this)-=rhs;} constexpr modint operator*(const modint &rhs) const noexcept {return modint(*this)*=rhs;} constexpr modint operator/(const modint &rhs) const noexcept {return modint(*this)/=rhs;} constexpr modint &operator+=(const modint &rhs) noexcept { v+=rhs.v; if (v>=mod) v-=mod; return *this; } constexpr modint &operator-=(const modint &rhs) noexcept { if (v0){ if (exp&1) res*=self; self*=self; exp>>=1; } return res; } template friend constexpr modint operator+(T x,modint y) noexcept {return modint(x)+y;} template friend constexpr modint operator-(T x,modint y) noexcept {return modint(x)-y;} template friend constexpr modint operator*(T x,modint y) noexcept {return modint(x)*y;} template friend constexpr modint operator/(T x,modint y) noexcept {return modint(x)/y;} constexpr modint &operator++() noexcept {return ++v,*this;} constexpr modint &operator--() noexcept {return --v,*this;} constexpr modint operator++(int) noexcept {modint t=*this; return ++v,t;} constexpr modint operator--(int) noexcept {modint t=*this; return --v,t;} constexpr bool operator==(const modint &rhs) const noexcept {return v==rhs.v;} constexpr bool operator!=(const modint &rhs) const noexcept {return v!=rhs.v;} constexpr bool operator!() const noexcept {return !v;} friend istream &operator>>(istream &s,modint &rhs) noexcept { i64 v; rhs=modint{(s>>v,v)}; return s; } friend ostream &operator<<(ostream &s,const modint &rhs) noexcept { return s< struct NumberTheoreticTransform{ using Mint=modint; vector roots; vector rev; int base,max_base; Mint root; NumberTheoreticTransform():base(1),rev{0,1},roots{Mint(0),Mint(1)}{ int tmp=mod-1; for (max_base=0;tmp%2==0;++max_base) tmp>>=1; root=2; while (root.pow((mod-1)>>1)==1) ++root; root=root.pow((mod-1)>>max_base); } void ensure_base(int nbase){ if (nbase<=base) return; rev.resize(1<>1]>>1)|((i&1)<<(nbase-1)); } roots.resize(1< &a){ const int n=a.size(); int zeros=__builtin_ctz(n); ensure_base(zeros); int shift=base-zeros; for (int i=0;i>shift)){ swap(a[i],a[rev[i]>>shift]); } } for (int k=1;k multiply(vector a,vector b){ int need=a.size()+b.size()-1; int nbase=1; while ((1< multiply(vector a,vector b){ vector A(a.size()),B(b.size()); for (int i=0;i C=multiply(A,B); vector res(C.size()); for (int i=0;i vector ArbitaryModConvolution(const vector &a,const vector &b){ int n=a.size(),m=b.size(); static constexpr int mod0=167772161,mod1=469762049,mod2=754974721; using mint0=modint; using mint1=modint; using mint2=modint; NumberTheoreticTransform ntt0; NumberTheoreticTransform ntt1; NumberTheoreticTransform ntt2; vector a0(n),b0(m); vector a1(n),b1(m); vector a2(n),b2(m); for (int i=0;i res(n+m-1); for (int i=0;i struct FormalPowerSeries:vector{ using vector::vector; using Poly=FormalPowerSeries; using MUL=function; static MUL &get_mul(){static MUL mul=nullptr; return mul;} static void set_mul(MUL f){get_mul()=f;} void shrink(){ while (this->size()&&this->back()==M(0)) this->pop_back(); } Poly pre(int deg) const {return Poly(this->begin(),this->begin()+min((int)this->size(),deg));} Poly operator+(const M &v) const {return Poly(*this)+=v;} Poly operator+(const Poly &p) const {return Poly(*this)+=p;} Poly operator-(const M &v) const {return Poly(*this)-=v;} Poly operator-(const Poly &p) const {return Poly(*this)-=p;} Poly operator*(const M &v) const {return Poly(*this)*=v;} Poly operator*(const Poly &p) const {return Poly(*this)*=p;} Poly operator/(const Poly &p) const {return Poly(*this)/=p;} Poly operator%(const Poly &p) const {return Poly(*this)%=p;} Poly &operator+=(const M &v){ if (this->empty()) this->resize(1); (*this)[0]+=v; return *this; } Poly &operator+=(const Poly &p){ if (p.size()>this->size()) this->resize(p.size()); for (int i=0;iempty()) this->resize(1); (*this)[0]-=v; return *this; } Poly &operator-=(const Poly &p){ if (p.size()>this->size()) this->resize(p.size()); for (int i=0;isize();++i) (*this)[i]*=v; return *this; } Poly &operator*=(const Poly &p){ if (this->empty()||p.empty()){ this->clear(); return *this; } assert(get_mul()!=nullptr); return *this=get_mul()(*this,p); } Poly &operator/=(const Poly &p){ if (this->size()clear(); return *this; } int n=this->size()-p.size()-1; return *this=(rev().pre(n)*p.rev().inv(n)).pre(n).rev(n); } Poly &operator%=(const Poly &p){return *this-=*this/p*p;} Poly operator<<(const int deg){ Poly res(*this); res.insert(res.begin(),deg,M(0)); return res; } Poly operator>>(const int deg){ if (this->size()<=deg) return {}; Poly res(*this); res.erase(res.begin(),res.begin()+deg); return res; } Poly operator-() const { Poly res(this->size()); for (int i=0;isize();++i) res[i]=-(*this)[i]; return res; } Poly rev(int deg=-1) const { Poly res(*this); if (~deg) res.resize(deg,M(0)); reverse(res.begin(),res.end()); return res; } Poly diff() const { Poly res(max(0,(int)this->size()-1)); for (int i=1;isize();++i) res[i-1]=(*this)[i]*M(i); return res; } Poly integral() const { Poly res(this->size()+1); res[0]=M(0); for (int i=0;isize();++i) res[i+1]=(*this)[i]/M(i+1); return res; } Poly inv(int deg=-1) const { assert((*this)[0]!=M(0)); if (deg<0) deg=this->size(); Poly res({M(1)/(*this)[0]}); for (int i=1;isize(); return (this->diff()*this->inv(deg)).pre(deg-1).integral(); } Poly sqrt(int deg=-1) const { assert((*this)[0]==M(1)); if (deg==-1) deg=this->size(); Poly res({M(1)}); M inv2=M(1)/M(2); for (int i=1;isize(); Poly res({M(1)}); for (int i=1;isize(); for (int i=0;isize();++i){ if ((*this)[i]==M(0)) continue; if (k*i>deg) return Poly(deg,M(0)); M inv=M(1)/(*this)[i]; Poly res=(((*this*inv)>>i).log()*k).exp()*(*this)[i].pow(k); res=(res<<(i*k)).pre(deg); if (res.size()0){ if (k&1) res=res*x%mod; x=x*x%mod; k>>=1; } return res; } }; using mint=modint; using FPS=FormalPowerSeries; const int MAX_N=100010; int main(){ cin.tie(0); ios::sync_with_stdio(false); auto mul=[&](const FPS::Poly &a,const FPS::Poly &b){ auto res=ArbitaryModConvolution(a,b); return FPS::Poly(res.begin(),res.end()); }; FPS::set_mul(mul); int K,N; cin >> K >> N; FPS a(MAX_N,0); a[0]+=1; for (;N--;){ int x; cin >> x; a[x]-=1; } a=a.inv(); cout << a[K] << '\n'; }