Language: C++

#include <string>
#include <vector>
#include<iostream>
#include<cstdio>
#include<cstdlib>
#include<stack>
#include<queue>
#include<cmath>
#include<algorithm>
#include<functional>
#include<list>
#include<deque>
#include<bitset>
#include<set>
#include<map>
#include<unordered_map>
#include<unordered_set>
#include<cstring>
#include<sstream>
#include<complex>
#include<iomanip>
#include<numeric>
#include<cassert>
#define X first
#define Y second
#define pb push_back
#define rep(X,Y) for (int (X) = 0;(X) < (Y);++(X))
#define reps(X,S,Y) for (int (X) = S;(X) < (Y);++(X))
#define rrep(X,Y) for (int (X) = (Y)-1;(X) >=0;--(X))
#define rreps(X,S,Y) for (int (X) = (Y)-1;(X) >= (S);--(X))
#define repe(X,Y) for ((X) = 0;(X) < (Y);++(X))
#define peat(X,Y) for (;(X) < (Y);++(X))
#define all(X) (X).begin(),(X).end()
#define rall(X) (X).rbegin(),(X).rend()
#define eb emplace_back
#define UNIQUE(X) (X).erase(unique(all(X)),(X).end())
#define Endl endl

using namespace std;
typedef long long ll;
typedef pair<int,int> pii;
typedef pair<ll,ll> pll;
template<class T> using vv=vector<vector<T>>;
template<class T> ostream& operator<<(ostream &os, const deque<T> &t) {
os<<"{"; rep(i,t.size()) {os<<t[i]<<",";} os<<"}"<<endl; return os;}
template<class T> ostream& operator<<(ostream &os, const vector<T> &t) {
os<<"{"; rep(i,t.size()) {os<<t[i]<<",";} os<<"}"<<endl; return os;}
template<class T,size_t n> ostream& operator<<(ostream &os, const array<T,n> &t) {
	os<<"{"; rep(i,n) {os<<t[i]<<",";} os<<"}"<<endl; return os;}
template<class S, class T> ostream& operator<<(ostream &os, const pair<S,T> &t) { return os<<"("<<t.first<<","<<t.second<<")";}
template<class S, class T,class U> ostream& operator<<(ostream &os, const tuple<S,T,U> &t) { return os<<"("<<get<0>(t)<<","<<get<1>(t)<<","<<get<2>(t)<<")";}
template<class S, class T,class U,class V> ostream& operator<<(ostream &os, const tuple<S,T,U,V> &t) { return os<<"("<<get<0>(t)<<","<<get<1>(t)<<","<<get<2>(t)<<","<<get<3>(t)<<")";}
template<class S, class T,class U,class V,class W> ostream& operator<<(ostream &os, const tuple<S,T,U,V,W> &t) { return os<<"("<<get<0>(t)<<","<<get<1>(t)<<","<<get<2>(t)<<","<<get<3>(t)<<","<<get<4>(t)<<")";}
template<class T> inline bool MX(T &l,const T &r){return l<r?l=r,1:0;}
template<class T> inline bool MN(T &l,const T &r){return l>r?l=r,1:0;}
//#undef NUIP
#ifdef NUIP
#define out(args...){vector<string> a_r_g_s=s_p_l_i_t(#args, ','); e_r_r(a_r_g_s.begin(), args); }
vector<string> s_p_l_i_t(const string &s, char c){vector<string> v;int d=0,f=0;string t;for(char c:s){if(!d&&c==',')v.pb(t),t="";else t+=c;if(c=='\"'||c=='\'')f^=1;if(!f&&c=='(')++d;if(!f&&c==')')--d;}v.pb(t);return move(v);}
void e_r_r(vector<string>::iterator it) {}
template<typename T, typename... Args> void e_r_r(vector<string>::iterator it, T a, Args... args){ if(*it==" 1"||*it=="1") cerr<<endl; else cerr << it -> substr((*it)[0] == ' ', it -> length()) << " = " << a << ", "; e_r_r(++it, args...);}
#else
#define out
#endif
#ifdef __cpp_init_captures
template<typename T>vector<T> table(int n, T v){ return vector<T>(n, v);}
template <class... Args> auto table(int n, Args... args){auto val = table(args...); return vector<decltype(val)>(n, move(val));}
#endif
const ll MOD=1e9+7;

ll modpow(ll r,ll n,ll m=MOD){
  ll re=1,d=r%m;
  if(n<0)(n%=m-1)+=m-1;
  for(;n;n/=2){
    if(n&1)(re*=d)%=m;
    (d*=d)%=m;
  }
  return re;
}
template <int mod=MOD> struct modInt{
  int v;
  modInt(ll v=0):v(v%MOD){}
  modInt(int v=0):v(v){}
  modInt operator+(const modInt &n)const{return v+n.v<mod ? v+n.v : v+n.v-mod;}
  modInt operator-(const modInt &n)const{return v-n.v<0 ? v-n.v+mod : v-n.v;}
  modInt operator*(const modInt &n)const{return ll(v)*n.v%mod;}
  modInt operator/(const modInt &n)const{return ll(v)*modpow(n.v%mod,-1,mod)%mod;}
  modInt operator+(const ll &n)const{return v+n<mod ? v+n : v+n-mod;}
  modInt operator-(const ll &n)const{return v-n<0 ? v-n+mod : v-n;}
  modInt operator*(const ll &n)const{return ll(v)*(n%mod)%mod;}
  modInt operator/(const ll &n)const{return ll(v)*modpow(n%mod,-1,mod)%mod;}
  modInt& operator+=(const modInt &n){v+=n.v; if(v>=mod) v-=mod; return *this;}
  modInt& operator-=(const modInt &n){v-=n.v; if(v<0) v+=mod; return *this;}
  modInt& operator*=(const modInt &n){v=ll(v)*n.v%mod; return *this;}
  modInt& operator/=(const modInt &n){v=ll(v)*modpow(n.v,-1,mod)%mod; return *this;}
  modInt& operator+=(const ll &n){v+=n; if(v>=mod) v-=mod; return *this;}
  modInt& operator-=(const ll &n){v-=n; if(v<0) v+=mod; return *this;}
  modInt& operator*=(const ll &n){v=ll(v)*n%mod; return *this;}
  modInt& operator/=(const ll &n){v=ll(v)*modpow(n,-1,mod)%mod; return *this;}
};
template<int mod> ostream& operator<<(ostream &os,const modInt<mod> &n){return os<<n.v;};
template<int mod> modInt<mod> operator+(const ll &n,const modInt<mod> &m){return m.v+n<mod ? m.v+n : m.v+n-mod;}
template<int mod> modInt<mod> operator-(const ll &n,const modInt<mod> &m){return n-m.v<0 ? n-m.v+mod : n-m.v;}
template<int mod> modInt<mod> operator*(const ll &n,const modInt<mod> &m){return ll(m.v)*(n%mod)%mod;}
template<int mod> modInt<mod> operator/(const ll &n,const modInt<mod> &m){return ll(m.v)*modpow(n%mod,-1,mod)%mod;}
typedef modInt<MOD> mint;
template <int mod> modInt<mod> modpow(modInt<mod> r,ll n){ modInt<mod> re(1); if(n<0)(n%=mod-1)+=mod-1; for(;n;n/=2){if(n&1) re*=r; r*=r;} return re;}
vector<mint> fact,finv,inv;
mint comb(ll n,ll r){ if(n<r||r<0)return 0; return fact[n]*finv[n-r]*finv[r];}
class Doralion{
  void Modinvs(vector<mint> &re,int n){ re.resize(n+1); re[1]=1; for(int i=2;i<=n;++i)re[i]=re[MOD%i]*(MOD-MOD/i);}
  void Facts(vector<mint> &re,int n){ re.resize(n+1); re[0]=1; rep(i,n)re[i+1]=re[i]*(i+1);}
  void Factinvs(vector<mint> &re,const vector<mint> &inv,int n){ re.resize(n+1); re[0]=1; rep(i,n)re[i+1]=re[i]*inv[i+1];}
public:
  Doralion(int n){ Modinvs(inv,n); Facts(fact,n); Factinvs(finv,inv,n);}
} doralion(21234567);


int main(){
  ios_base::sync_with_stdio(false);
  cout<<fixed<<setprecision(0);
	int H,W,t;
	cin>>H>>W>>t;
	// if(t==1){
	// 	cout<<4<<endl;
	// 	return 0;
	// }
	mint re=0;
	vector<mint> poyo(H+W+10);
	reps(i,1,H+W+10) poyo[i]=poyo[i-1]+i*(i-1ll)%MOD;
	rep(h,H+1){
		int w=t-h;
		if(w<0 || w>W) continue;
		mint sum=0;
		// rep(s,h+w+1){
		// 	// rep(a,min(s,h)+1) sum+=s*comb(s-1,a)*comb(h+w-s,h-a);
		// 	// if(s&&h+w) sum+=s*(s-1ll)%MOD*comb(h+w-2,h-1);
		// 	// sum+=s*comb(h+w-1,h-1);
		// 	// sum+=s*comb(h+w-1,h);
		// 	// sum+=comb(h+w,h);
		// }
		if(h) sum+=poyo[h+w]*comb(h+w-2,h-1);
		mint tmp=(h+w)*(h+w+1ll)/2;
		if(h) sum+=tmp*comb(h+w-1,h-1);
		sum+=tmp*comb(h+w-1,h);
		sum+=(h+w+1)*comb(h+w,h);
		// reps(a,1,h+1)reps(b,1,w+1) sum+=(a+b)*(a+b-1ll)%MOD*comb(a+b-2,a-1)*comb(h-a+w-b,h-a);
		// reps(a,1,h+1)rep(b,w+1) sum+=(a+b)*comb(a+b-1,a-1)*comb(h-a+w-b,h-a);
		// rep(a,h+1)reps(b,1,w+1) sum+=(a+b)*comb(a+b-1,a)*comb(h-a+w-b,h-a);
		// rep(a,h+1)rep(b,w+1) sum+=comb(a+b,a)*comb(h-a+w-b,h-a);
		sum-=comb(h+w,h);
		re+=sum*fact[H]*finv[H-h]*fact[W]*finv[W-w];
		out(h,w,re,1);
	}
	cout<<re<<endl;
  return 0;
}
	// rep(h,H+1){
	// 	int w=t-h;
	// 	if(w<0 || w>W) continue;
	// 	rep(a,h+1)rep(b,w+1) re+=(a+1ll)*(b+1ll)%MOD*comb(a+b,a)*comb(h-a+w-b,h-a)
	// 		*fact[H]*finv[H-h]*fact[W]*finv[W-w];
	// 	re-=comb(h+w,h)*fact[H]*finv[H-h]*fact[W]*finv[W-w];
	// 	out(h,w,re,1);
	// }


Compiler output:
In file included from /usr/include/c++/14/bits/alloc_traits.h:33,
                 from /usr/include/c++/14/ext/alloc_traits.h:34,
                 from /usr/include/c++/14/bits/basic_string.h:39,
                 from /usr/include/c++/14/string:54,
                 from a.cc:1:
/usr/include/c++/14/bits/stl_construct.h: In instantiation of 'void std::_Construct(_Tp*, _Args&& ...) [with _Tp = modInt<1000000007>; _Args = {}]':
/usr/include/c++/14/bits/stl_uninitialized.h:643:18:   required from 'static _ForwardIterator std::__uninitialized_default_n_1<_TrivialValueType>::__uninit_default_n(_ForwardIterator, _Size) [with _ForwardIterator = modInt<1000000007>*; _Size = long unsigned int; bool _TrivialValueType = false]'
  643 |                 std::_Construct(std::__addressof(*__cur));
      |                 ~~~~~~~~~~~~~~~^~~~~~~~~~~~~~~~~~~~~~~~~~
/usr/include/c++/14/bits/stl_uninitialized.h:712:20:   required from '_ForwardIterator std::__uninitialized_default_n(_ForwardIterator, _Size) [with _ForwardIterator = modInt<1000000007>*; _Size = long unsigned int]'
  710 |       return __uninitialized_default_n_1<__is_trivial(_ValueType)
      |              ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
  711 |                                          && __can_fill>::
      |                                          ~~~~~~~~~~~~~~~~
  712 |         __uninit_default_n(__first, __n);
      |         ~~~~~~~~~~~~~~~~~~^~~~~~~~~~~~~~
/usr/include/c++/14/bits/stl_uninitialized.h:779:44:   required from '_ForwardIterator std::__uninitialized_default_n_a(_ForwardIterator, _Size, allocator<_Tp>&) [with _ForwardIterator = modInt<1000000007>*; _Size = long unsigned int; _Tp = modInt<1000000007>]'
  779 |     { return std::__uninitialized_default_n(__first, __n); }
      |              ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~^~~~~~~~~~~~~~
/usr/include/c++/14/bits/vector.tcc:821:35:   required from 'void std::vector<_Tp, _Alloc>::_M_default_append(size_type) [with _Tp = modInt<1000000007>; _Alloc = std::allocator<modInt<1000000007> >; size_type = long unsigned int]'
  821 |                 std::__uninitialized_default_n_a(this->_M_impl._M_finish,
      |                 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~^~~~~~~~~~~~~~~~~~~~~~~~~
  822 |                                                  __n, _M_get_Tp_allocator());
      |                                                  ~~~~~~~~~~~~~~~~~~~~~~~~~~~
/usr/include/c++/14/bits/stl_vector.h:1016:4:   required from 'void std::vector<_Tp, _Alloc>::resize(size_type) [with _Tp = modInt<1000000007>; _Alloc = std::allocator<modInt<1000000007> >; size_type = long unsigned int]'
 1016 |           _M_default_append(__new_size - size());
      |           ^~~~~~~~~~~~~~~~~
a.cc:111:50:   required from here
  111 |   void Modinvs(vector<mint> &re,int n){ re.resize(n+1); re[1]=1; for(int i=2;i<=n;++i)re[i]=re[MOD%i]*(MOD-MOD/i);}
      |                                         ~~~~~~~~~^~~~~
/usr/include/c++/14/bits/stl_construct.h:119:7: error: call of overloaded 'modInt()' is ambiguous
  119 |       ::new((void*)__p) _Tp(std::forward<_Args>(__args)...);
      |       ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
a.cc:83:3: note: candidate: 'modInt<mod>::modInt(int) [with int mod = 1000000007]'
   83 |   modInt(int v=0):v(v){}
      |   ^~~~~~
a.cc:82:3: note: candidate: 'modInt<mod>::modInt(ll) [with int mod = 1000000007; ll = long long int]'
   82 |   modInt(ll v=0):v(v%MOD){}
      |   ^~~~~~