Problem:
F: Steps
問題
AORイカちゃんは株式会社AORの視察に訪れた。
株式会社AORは $N$ 階建てのビルであり、地下階は存在しない。
AORイカちゃんは $1$ 階から視察を開始し、ちょうど $M$ 回移動することにした。
$1$ 回の移動で $i$ 階から $i + 1$ 階、あるいは $i - 1$ 階に移動することができる。
ただし、 移動先が $0$ 階や $N + 1$ 階となるように移動することはできない。
AORイカちゃんが全ての階に一度以上訪れる移動方法の通り数を $1000000007 (= 10^9 + 7)$ で割った余りを求めよ。
なお、 $1$ 階にはすでに訪れたものとする。
制約
$2 \leq N \leq 10^5$
$1 \leq M \leq 10^5$
入力は全て整数で与えらえる。
入力形式
入力は以下の形式で与えられる。
$N \ M$
出力
移動方法の通り数を $1000000007 (= 10 ^ 9 + 7)$ で割った余りを出力せよ。また、末尾に改行も出力せよ。
サンプル
サンプル入力 1
3 5
サンプル出力 1
3
サンプル入力 2
3 1
サンプル出力 2
0
サンプル入力 3
4883 5989
サンプル出力 3
956662807

Submitted solutions:

code_snippet: //#define _GLIBCXX_DEBUG

#include<bits/stdc++.h>
using namespace std;

#define endl '\n'
#define lfs cout<<fixed<<setprecision(10)
#define ALL(a)  (a).begin(),(a).end()
#define ALLR(a)  (a).rbegin(),(a).rend()
#define UNIQUE(a) (a).erase(unique((a).begin(),(a).end()),(a).end())
#define spa << " " <<
#define fi first
#define se second
#define MP make_pair
#define MT make_tuple
#define PB push_back
#define EB emplace_back
#define rep(i,n,m) for(ll i = (n); i < (ll)(m); i++)
#define rrep(i,n,m) for(ll i = (ll)(m) - 1; i >= (ll)(n); i--)
using ll = long long;
using ld = long double;
const ll MOD1 = 1e9+7;
const ll MOD9 = 998244353;
const ll INF = 1e18;
using P = pair<ll, ll>;
template<typename T> using PQ = priority_queue<T>;
template<typename T> using QP = priority_queue<T,vector<T>,greater<T>>;
template<typename T1, typename T2>bool chmin(T1 &a,T2 b){if(a>b){a=b;return true;}else return false;}
template<typename T1, typename T2>bool chmax(T1 &a,T2 b){if(a<b){a=b;return true;}else return false;}
ll median(ll a,ll b, ll c){return a+b+c-max({a,b,c})-min({a,b,c});}
void ans1(bool x){if(x) cout<<"Yes"<<endl;else cout<<"No"<<endl;}
void ans2(bool x){if(x) cout<<"YES"<<endl;else cout<<"NO"<<endl;}
void ans3(bool x){if(x) cout<<"Yay!"<<endl;else cout<<":("<<endl;}
template<typename T1,typename T2>void ans(bool x,T1 y,T2 z){if(x)cout<<y<<endl;else cout<<z<<endl;}  
template<typename T1,typename T2,typename T3>void anss(T1 x,T2 y,T3 z){ans(x!=y,x,z);};  
template<typename T>void debug(const T &v,ll h,ll w,string sv=" "){for(ll i=0;i<h;i++){cout<<v[i][0];for(ll j=1;j<w;j++)cout<<sv<<v[i][j];cout<<endl;}};
template<typename T>void debug(const T &v,ll n,string sv=" "){if(n!=0)cout<<v[0];for(ll i=1;i<n;i++)cout<<sv<<v[i];cout<<endl;};
template<typename T>void debug(const vector<T>&v){debug(v,v.size());}
template<typename T>void debug(const vector<vector<T>>&v){for(auto &vv:v)debug(vv,vv.size());}
template<typename T>void debug(stack<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(queue<T> st){while(!st.empty()){cout<<st.front()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(deque<T> st){while(!st.empty()){cout<<st.front()<<" ";st.pop_front();}cout<<endl;}
template<typename T>void debug(PQ<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(QP<T> st){while(!st.empty()){cout<<st.top()<<" ";st.pop();}cout<<endl;}
template<typename T>void debug(const set<T>&v){for(auto z:v)cout<<z<<" ";cout<<endl;}
template<typename T>void debug(const multiset<T>&v){for(auto z:v)cout<<z<<" ";cout<<endl;}
template<typename T,size_t size>void debug(const array<T, size> &a){for(auto z:a)cout<<z<<" ";cout<<endl;}
template<typename T,typename V>void debug(const map<T,V>&v){for(auto z:v)cout<<"["<<z.first<<"]="<<z.second<<",";cout<<endl;}
template<typename T>vector<vector<T>>vec(ll x, ll y, T w){vector<vector<T>>v(x,vector<T>(y,w));return v;}
ll gcd(ll x,ll y){ll r;while(y!=0&&(r=x%y)!=0){x=y;y=r;}return y==0?x:y;}
vector<ll>dx={1,-1,0,0,1,1,-1,-1};vector<ll>dy={0,0,1,-1,1,-1,1,-1};
template<typename T>vector<T> make_v(size_t a,T b){return vector<T>(a,b);}
template<typename... Ts>auto make_v(size_t a,Ts... ts){return vector<decltype(make_v(ts...))>(a,make_v(ts...));}
template<typename T1, typename T2>ostream &operator<<(ostream &os, const pair<T1, T2>&p){return os << p.first << " " << p.second;}
template<typename T>ostream &operator<<(ostream &os, const vector<T> &v){for(auto &z:v)os << z << " ";cout<<"|"; return os;}
template<typename T>void rearrange(vector<int>&ord, vector<T>&v){
  auto tmp = v;
  for(int i=0;i<tmp.size();i++)v[i] = tmp[ord[i]];
}
template<typename Head, typename... Tail>void rearrange(vector<int>&ord,Head&& head, Tail&&... tail){
  rearrange(ord, head);
  rearrange(ord, tail...);
}
template<typename T> vector<int> ascend(const vector<T>&v){
  vector<int>ord(v.size());iota(ord.begin(),ord.end(),0);
  sort(ord.begin(),ord.end(),[&](int i,int j){return v[i]<v[j];});
  return ord;
}
template<typename T> vector<int> descend(const vector<T>&v){
  vector<int>ord(v.size());iota(ord.begin(),ord.end(),0);
  sort(ord.begin(),ord.end(),[&](int i,int j){return v[i]>v[j];});
  return ord;
}
ll FLOOR(ll n,ll div){assert(div>0);return n>=0?n/div:(n-div+1)/div;}
ll CEIL(ll n,ll div){assert(div>0);return n>=0?(n+div-1)/div:n/div;}
ll digitsum(ll n){ll ret=0;while(n){ret+=n%10;n/=10;}return ret;}
ll modulo(ll n,ll d){return (n%d+d)%d;};
template<typename T>T min(const vector<T>&v){return *min_element(v.begin(),v.end());}
template<typename T>T max(const vector<T>&v){return *max_element(v.begin(),v.end());}
template<typename T>T acc(const vector<T>&v){return accumulate(v.begin(),v.end(),T(0));};
template<typename T>T reverse(const T &v){return T(v.rbegin(),v.rend());};
//mt19937 mt(chrono::steady_clock::now().time_since_epoch().count());
int popcount(ll x){return __builtin_popcountll(x);};
int poplow(ll x){return __builtin_ctzll(x);};
int pophigh(ll x){return 63 - __builtin_clzll(x);};
template<typename T>T poll(queue<T> &q){auto ret=q.front();q.pop();return ret;};
template<typename T>T poll(priority_queue<T> &q){auto ret=q.top();q.pop();return ret;};
template<typename T>T poll(QP<T> &q){auto ret=q.top();q.pop();return ret;};
template<typename T>T poll(stack<T> &s){auto ret=s.top();s.pop();return ret;};
ll MULT(ll x,ll y){if(LLONG_MAX/x<=y)return LLONG_MAX;return x*y;}
ll POW2(ll x, ll k){ll ret=1,mul=x;while(k){if(mul==LLONG_MAX)return LLONG_MAX;if(k&1)ret=MULT(ret,mul);mul=MULT(mul,mul);k>>=1;}return ret;}
ll POW(ll x, ll k){ll ret=1;for(int i=0;i<k;i++){if(LLONG_MAX/x<=ret)return LLONG_MAX;ret*=x;}return ret;}
template< typename T = int >
struct edge {
  int to;
  T cost;
  int id;
  edge():id(-1){};
  edge(int to, T cost = 1, int id = -1):to(to), cost(cost), id(id){}
  operator int() const { return to; }
};

template<typename T>
using Graph = vector<vector<edge<T>>>;
template<typename T>
Graph<T>revgraph(const Graph<T> &g){
  Graph<T>ret(g.size());
  for(int i=0;i<g.size();i++){
    for(auto e:g[i]){
      int to = e.to;
      e.to = i;
      ret[to].push_back(e);
    }
  }
  return ret;
}
template<typename T>
Graph<T> readGraph(int n,int m,int indexed=1,bool directed=false,bool weighted=false){
  Graph<T> ret(n);
  for(int es = 0; es < m; es++){
    int u,v;
    T w=1;
    cin>>u>>v;u-=indexed,v-=indexed;
    if(weighted)cin>>w;
    ret[u].emplace_back(v,w,es);
    if(!directed)ret[v].emplace_back(u,w,es);
  }
  return ret;
}
template<typename T>
Graph<T> readParent(int n,int indexed=1,bool directed=true){
  Graph<T>ret(n);
  for(int i=1;i<n;i++){
    int p;cin>>p;
    p-=indexed;
    ret[p].emplace_back(i);
    if(!directed)ret[i].emplace_back(p);
  }
  return ret;
}
template< int mod >
struct ModInt {
  int x;

  ModInt() : x(0) {}

  ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}

  ModInt &operator+=(const ModInt &p) {
    if((x += p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator-=(const ModInt &p) {
    if((x += mod - p.x) >= mod) x -= mod;
    return *this;
  }

  ModInt &operator*=(const ModInt &p) {
    x = (int) (1LL * x * p.x % mod);
    return *this;
  }

  ModInt &operator/=(const ModInt &p) {
    *this *= p.inverse();
    return *this;
  }

  ModInt operator-() const { return ModInt(-x); }

  friend ModInt operator+(const ModInt& lhs, const ModInt& rhs) {
        return ModInt(lhs) += rhs;
  }
  friend ModInt operator-(const ModInt& lhs, const ModInt& rhs) {
        return ModInt(lhs) -= rhs;
  }
  friend ModInt operator*(const ModInt& lhs, const ModInt& rhs) {
        return ModInt(lhs) *= rhs;
  }
  friend ModInt operator/(const ModInt& lhs, const ModInt& rhs) {
        return ModInt(lhs) /= rhs;
  }

  bool operator==(const ModInt &p) const { return x == p.x; }

  bool operator!=(const ModInt &p) const { return x != p.x; }

  ModInt inverse() const {
    int a = x, b = mod, u = 1, v = 0, t;
    while(b > 0) {
      t = a / b;
      swap(a -= t * b, b);
      swap(u -= t * v, v);
    }
    return ModInt(u);
  }

  ModInt pow(int64_t n) const {
    ModInt ret(1), mul(x);
    while(n > 0) {
      if(n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
  friend ostream &operator<<(ostream &os, const ModInt &p) {
    return os << p.x;
  }

  friend istream &operator>>(istream &is, ModInt &a) {
    int64_t t;
    is >> t;
    a = ModInt< mod >(t);
    return (is);
  }

  static int get_mod() { return mod; }
};

template< typename T >
struct Combination {
  vector< T > _fact, _rfact, _inv;

  Combination(ll sz) : _fact(sz + 1), _rfact(sz + 1), _inv(sz + 1) {
    _fact[0] = _rfact[sz] = _inv[0] = 1;
    for(ll i = 1; i <= sz; i++) _fact[i] = _fact[i - 1] * i;
    _rfact[sz] /= _fact[sz];
    for(ll i = sz - 1; i >= 0; i--) _rfact[i] = _rfact[i + 1] * (i + 1);
    for(ll i = 1; i <= sz; i++) _inv[i] = _rfact[i] * _fact[i - 1];
  }

  inline T fact(ll k) const { return _fact[k]; }

  inline T rfact(ll k) const { return _rfact[k]; }

  inline T inv(ll k) const { return _inv[k]; }

  T P(ll n, ll r) const {
    if(r < 0 || n < r) return 0;
    return fact(n) * rfact(n - r);
  }

  T C(ll p, ll q) const {
    if(q < 0 || p < q) return 0;
    return fact(p) * rfact(q) * rfact(p - q);
  }
  
  T RC(ll p, ll q) const {
    if(q < 0 || p < q) return 0;
    return rfact(p) * fact(q) * fact(p - q);
  }

  T H(ll n, ll r) const {
    if(n < 0 || r < 0) return (0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }
};
using modint = ModInt< MOD1 >;modint pow(ll n, ll x){return modint(n).pow(x);}modint pow(modint n, ll x){return n.pow(x);}
//using modint=ld;
using Comb=Combination<modint>;
int main(){
  cin.tie(nullptr);
  ios_base::sync_with_stdio(false);
  ll res=0,buf=0;
  bool judge = true;
  ll N,m;cin>>N>>m;
  Comb comb(400005);
  auto calc=[&](ll n){
    modint ret=0;
    rep(i,0,m+1){
      modint add=comb.C(m,i);
      ll px=modulo((1+i-(m-i)),2*(n+1));
      if(px>=1&&px<=n)ret+=add;
      ll py=modulo((2*n+1+i-(m-i)),2*(n+1));
      if(py>=1&&py<=n)ret-=add;
    }
    return ret;
  };
  modint p=calc(N);
  modint q=calc(N-1);
  cout<<p-q<<endl;
  return 0;
}
accuracy: 1
time_ms: 10
memory_kb: 7848
final_rank: 1

code_snippet: #include <bits/stdc++.h>
using namespace std;

template <int mod = (int)(1e9 + 7)>
struct ModInt {
  int x;
  ModInt() : x(0) {}
  ModInt(int64_t y) : x(y >= 0 ? y % mod : (mod - (-y) % mod) % mod) {}
  ModInt &operator+=(const ModInt &p) {
    if ((x += p.x) >= mod) x -= mod;
    return *this;
  }
  ModInt &operator-=(const ModInt &p) {
    if ((x += mod - p.x) >= mod) x -= mod;
    return *this;
  }
  ModInt &operator*=(const ModInt &p) {
    x = (int)(1LL * x * p.x % mod);
    return *this;
  }
  ModInt &operator/=(const ModInt &p) {
    *this *= p.inverse();
    return *this;
  }
  ModInt operator-() const { return ModInt(-x); }
  ModInt operator+(const ModInt &p) const { return ModInt(*this) += p; }
  ModInt operator-(const ModInt &p) const { return ModInt(*this) -= p; }
  ModInt operator*(const ModInt &p) const { return ModInt(*this) *= p; }
  ModInt operator/(const ModInt &p) const { return ModInt(*this) /= p; }
  bool operator==(const ModInt &p) const { return x == p.x; }
  bool operator!=(const ModInt &p) const { return x != p.x; }
  ModInt inverse() const {
    int a = x, b = mod, u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b);
      swap(u -= t * v, v);
    }
    return ModInt(u);
  }
  ModInt pow(int64_t n) const {
    ModInt res(1), mul(x);
    while (n) {
      if (n & 1) res *= mul;
      mul *= mul;
      n >>= 1;
    }
    return res;
  }
  friend ostream &operator<<(ostream &os, const ModInt &p) { return os << p.x; }
  friend istream &operator>>(istream &is, ModInt &a) {
    int64_t t;
    is >> t;
    a = ModInt<mod>(t);
    return (is);
  }
  static int get_mod() { return mod; }
};

struct Combination {
  vector<ModInt<>> _fact, _rfact, _inv;
  Combination(long long nsize = 5000000)
      : _fact(nsize + 1), _rfact(nsize + 1), _inv(nsize + 1) {
    _fact[0] = _rfact[nsize] = _inv[0] = 1;
    for (int i = 1; i <= nsize; i++) _fact[i] = _fact[i - 1] * i;
    _rfact[nsize] /= _fact[nsize];
    for (int i = nsize - 1; i >= 0; i--) _rfact[i] = _rfact[i + 1] * (i + 1);
    for (int i = 1; i <= nsize; i++) _inv[i] = _rfact[i] * _fact[i - 1];
  }
  inline ModInt<> fact(int k) const { return _fact[k]; }

  inline ModInt<> rfact(int k) const { return _rfact[k]; }

  inline ModInt<> inv(int k) const { return _inv[k]; }

  ModInt<> P(int n, int r) const {
    if (r < 0 || n < r) return 0;
    return fact(n) * rfact(n - r);
  }

  ModInt<> C(int p, int q) const {
    if (q < 0 || p < q) return 0;
    return fact(p) * rfact(q) * rfact(p - q);
  }

  ModInt<> largeC(long long p, long long q) const {
    if (q < 0 || p < q) return 0;
    ModInt<> res = rfact(q);
    for (int i = 0; i < q; ++i) res *= p - i;
    return res;
  }

  // n types,choose r
  ModInt<> H(int n, int r) const {
    if (n < 0 || r < 0) return (0);
    return r == 0 ? 1 : C(n + r - 1, r);
  }

  ModInt<> Catalan(int n) {
    // C(2n,n) / (n + 1)
    return fact(2 * n) * rfact(n + 1) * rfact(n);
  }
};
using mint = ModInt<>;

int m;
Combination com;

mint solve(int n);

int main() {
  int n;
  cin >> n >> m;
  cout << solve(n) - solve(n - 1) << endl;
  return 0;
}

mint solve(int n) {
  mint res;
  for (int i = 0; i < n; ++i)
    if (((m - i) ^ 1) & 1 && m >= i) {
      int up = (m + i) / 2, sig = -1, now = up, a = -n, b = -1;
      mint tmp = com.C(m, up);
      res += com.C(m, up);
      while (now + a >= 0) {
        now += a;
        tmp += com.C(m, now) * sig;
        res += com.C(m, now) * sig;
        sig *= -1;
        swap(a, b);
      }
      now = up;
      a = 1, b = n;
      sig = -1;
      while (now + a <= m) {
        now += a;
        tmp += com.C(m, now) * sig;
        res += com.C(m, now) * sig;
        sig *= -1;
        swap(a, b);
      }
    }
  return res;
}
accuracy: 1
time_ms: 50
memory_kb: 61684
final_rank: 4

code_snippet: #include <bits/stdc++.h>
using namespace std;
#define REP(i, n) for (long long i = 0, max_i = (n); i < max_i; i++)
template <class T>
ostream& operator<<(ostream& s, vector<T>& d) {
    REP(i, d.size()) s << d[i] << (i == d.size() - 1 ? "" : " ");
    return s;
}
using Int = long long;

template <typename T>
T pow(T a, long long n, T e = 1) {
    T ret = e;
    while (n) {
        if (n & 1) ret *= a;
        a *= a;
        n >>= 1;
    }
    return ret;
}

template <int mod>
struct ModInt {
    int x;
    ModInt() : x(0) {}
    ModInt(long long x_) {
        if ((x = x_ % mod + mod) >= mod) x -= mod;
    }
    ModInt& operator+=(ModInt rhs) {
        if ((x += rhs.x) >= mod) x -= mod;
        return *this;
    }
    ModInt& operator-=(ModInt rhs) {
        if ((x -= rhs.x) < 0) x += mod;
        return *this;
    }
    ModInt& operator*=(ModInt rhs) {
        x = (unsigned long long)x * rhs.x % mod;
        return *this;
    }
    ModInt& operator/=(ModInt rhs) {
        x = (unsigned long long)x * rhs.inv().x % mod;
        return *this;
    }

    ModInt operator-() const { return -x < 0 ? mod - x : -x; }
    ModInt operator+(ModInt rhs) const { return ModInt(*this) += rhs; }
    ModInt operator-(ModInt rhs) const { return ModInt(*this) -= rhs; }
    ModInt operator*(ModInt rhs) const { return ModInt(*this) *= rhs; }
    ModInt operator/(ModInt rhs) const { return ModInt(*this) /= rhs; }
    bool operator==(ModInt rhs) const { return x == rhs.x; }
    bool operator!=(ModInt rhs) const { return x != rhs.x; }
    ModInt inv() const { return pow(*this, mod - 2); }

    friend ostream& operator<<(ostream& s, ModInt<mod> a) {
        s << a.x;
        return s;
    }
    friend istream& operator>>(istream& s, ModInt<mod>& a) {
        s >> a.x;
        return s;
    }
};

using mint = ModInt<1000000007>;

constexpr int MAX_FACTORIAL = 202020;

mint fact[MAX_FACTORIAL];
mint factinv[MAX_FACTORIAL];

void init() {
    fact[0] = 1;
    for (int i = 1; i < MAX_FACTORIAL; i++) {
        fact[i] = fact[i - 1] * i;
    }
    factinv[MAX_FACTORIAL - 1] = fact[MAX_FACTORIAL - 1].inv();
    for (int i = MAX_FACTORIAL - 2; i >= 0; i--) {
        factinv[i] = factinv[i + 1] * (i + 1);
    }
    assert(factinv[0] == 1);
}

mint ncr(int n, int r) {
    if (r < 0 || r > n) return 0;
    return fact[n] * factinv[r] * factinv[n - r];
}

// https://arxiv.org/pdf/1407.2197.pdf
mint D(int a, int b, int s, int t) {
    if (a + t < b || b < a + s) return 0;
    mint ret = 0;
    int l = -1e9, r = 0;
    while (l < r - 1) {
        int m = (l + r) / 2;
        long long p = a - 1LL * m * (t - s + 2);
        long long q = a - 1LL * m * (t - s + 2) + t + 1;
        if (0 <= p && p <= a + b || 0 <= q && q <= a + b) {
            r = m;
        } else {
            l = m;
        }
    }
    int L = 0, R = 1e9;
    while (L < R - 1) {
        int m = (L + R) / 2;
        long long p = a - 1LL * m * (t - s + 2);
        long long q = a - 1LL * m * (t - s + 2) + t + 1;
        if (0 <= p && p <= a + b || 0 <= q && q <= a + b) {
            L = m;
        } else {
            R = m;
        }
    }
    for (int k = r; k <= L; k++) {
        mint f = ncr(a + b, a - k * (t - s + 2));
        f -= ncr(a + b, a - k * (t - s + 2) + t + 1);
        ret += f;
    }
    return ret;
}

// [0, n]のマス目内をm回移動する場合の数でスタート位置が0
mint ext_catalan(int n, int m) {
    if (n == 0) return 0;
    mint ret = 0;
    for (int i = 0; i <= m; i++) {
        ret += D(i, m - i, 0, n);
    }
    return ret;
}

mint solve(int n, int m) { return ext_catalan(n, m) - ext_catalan(n - 1, m); }

int main() {
    init();
    int n, m;
    cin >> n >> m;
    cout << solve(n - 1, m) << endl;
}
accuracy: 1
time_ms: 20
memory_kb: 4676
final_rank: 2

code_snippet: #include <bits/stdc++.h>
using namespace std;
#define REP(i, n) for (long long i = 0, max_i = (n); i < max_i; i++)
template <class T>
ostream& operator<<(ostream& s, vector<T>& d) {
    REP(i, d.size()) s << d[i] << (i == d.size() - 1 ? "" : " ");
    return s;
}
using Int = long long;

template <typename T>
T pow(T a, long long n, T e = 1) {
    T ret = e;
    while (n) {
        if (n & 1) ret *= a;
        a *= a;
        n >>= 1;
    }
    return ret;
}

template <int mod>
struct ModInt {
    int x;
    ModInt() : x(0) {}
    ModInt(long long x_) {
        if ((x = x_ % mod + mod) >= mod) x -= mod;
    }
    ModInt& operator+=(ModInt rhs) {
        if ((x += rhs.x) >= mod) x -= mod;
        return *this;
    }
    ModInt& operator-=(ModInt rhs) {
        if ((x -= rhs.x) < 0) x += mod;
        return *this;
    }
    ModInt& operator*=(ModInt rhs) {
        x = (unsigned long long)x * rhs.x % mod;
        return *this;
    }
    ModInt& operator/=(ModInt rhs) {
        x = (unsigned long long)x * rhs.inv().x % mod;
        return *this;
    }

    ModInt operator-() const { return -x < 0 ? mod - x : -x; }
    ModInt operator+(ModInt rhs) const { return ModInt(*this) += rhs; }
    ModInt operator-(ModInt rhs) const { return ModInt(*this) -= rhs; }
    ModInt operator*(ModInt rhs) const { return ModInt(*this) *= rhs; }
    ModInt operator/(ModInt rhs) const { return ModInt(*this) /= rhs; }
    bool operator==(ModInt rhs) const { return x == rhs.x; }
    bool operator!=(ModInt rhs) const { return x != rhs.x; }
    ModInt inv() const { return pow(*this, mod - 2); }

    friend ostream& operator<<(ostream& s, ModInt<mod> a) {
        s << a.x;
        return s;
    }
    friend istream& operator>>(istream& s, ModInt<mod>& a) {
        s >> a.x;
        return s;
    }
};

using mint = ModInt<1000000007>;

constexpr int MAX_FACTORIAL = 202020;

mint fact[MAX_FACTORIAL];
mint factinv[MAX_FACTORIAL];

void init() {
    fact[0] = 1;
    for (int i = 1; i < MAX_FACTORIAL; i++) {
        fact[i] = fact[i - 1] * i;
    }
    factinv[MAX_FACTORIAL - 1] = fact[MAX_FACTORIAL - 1].inv();
    for (int i = MAX_FACTORIAL - 2; i >= 0; i--) {
        factinv[i] = factinv[i + 1] * (i + 1);
    }
    assert(factinv[0] == 1);
}

mint ncr(int n, int r) {
    if (r < 0 || r > n) return 0;
    return fact[n] * factinv[r] * factinv[n - r];
}

// https://arxiv.org/pdf/1407.2197.pdf
mint D(int a, int b, int s, int t) {
    if (a + t < b || b < a + s) return 0;
    mint ret = 0;
    int l = -1e9, r = 0;
    while (l < r - 1) {
        int m = (l + r) / 2;
        long long p = a - 1LL * m * (t - s + 2);
        long long q = a - 1LL * m * (t - s + 2) + t + 1;
        if (0 <= p && p <= a + b || 0 <= q && q <= a + b) {
            r = m;
        } else {
            l = m;
        }
    }
    int L = 0, R = 1e9;
    while (L < R - 1) {
        int m = (L + R) / 2;
        long long p = a - 1LL * m * (t - s + 2);
        long long q = a - 1LL * m * (t - s + 2) + t + 1;
        if (0 <= p && p <= a + b || 0 <= q && q <= a + b) {
            L = m;
        } else {
            R = m;
        }
    }
    for (int k = r; k <= L; k++) {
        mint f = ncr(a + b, a - k * (t - s + 2));
        f -= ncr(a + b, a - k * (t - s + 2) + t + 1);
        ret += f;
    }
    return ret;
}

mint hoge(int n, int m) {
    if (n == 0) return 0;
    mint ret = 0;
    for (int i = 0; i <= m; i++) {
        ret += D(i, m - i, 0, n);
    }
    return ret;
}

mint solve(int n, int m) { return hoge(n, m) - hoge(n - 1, m); }

int main() {
    init();
    int n, m;
    cin >> n >> m;
    cout << solve(n - 1, m) << endl;
}
accuracy: 1
time_ms: 20
memory_kb: 4676
final_rank: 2

code_snippet: #include <bits/stdc++.h>
using namespace std;

template<int MOD>
struct mod_int {
	static const int mod = MOD;
	unsigned x;
	mod_int() : x(0) { }
	mod_int(int sig) { int sigt = sig % MOD; if (sigt < 0) sigt += MOD; x = sigt; }
	mod_int(long long sig) { int sigt = sig % MOD; if (sigt < 0) sigt += MOD; x = sigt; }
	int get() const { return (int)x; }

	mod_int &operator+=(mod_int that) { if ((x += that.x) >= MOD) x -= MOD; return *this; }
	mod_int &operator-=(mod_int that) { if ((x += MOD - that.x) >= MOD) x -= MOD; return *this; }
	mod_int &operator*=(mod_int that) { x = (unsigned long long)x * that.x % MOD; return *this; }
	mod_int &operator/=(mod_int that) { return *this *= that.inverse(); }

	mod_int operator+(mod_int that) const { return mod_int(*this) += that; }
	mod_int operator-(mod_int that) const { return mod_int(*this) -= that; }
	mod_int operator*(mod_int that) const { return mod_int(*this) *= that; }
	mod_int operator/(mod_int that) const { return mod_int(*this) /= that; }

	mod_int inverse() const {
		long long a = x, b = MOD, u = 1, v = 0;
		while (b) {
			long long t = a / b;
			a -= t * b; swap(a, b);
			u -= t * v; swap(u, v);
		}
		return mod_int(u);
	}
};

template<int MOD>
istream& operator >> (istream& is, mod_int<MOD>& val) {
	long long x;
	is >> x; val = x;
	return is;
}

template<int MOD>
ostream& operator << (ostream& os, const mod_int<MOD>& val) {
	os << val.get();
	return os;
}

using mint = mod_int<1000000007>;

const int MAX = 2e6;

bool inited = false;
mint fac[MAX + 1];
mint rfac[MAX + 1];

void init() {
	inited = true;
	fac[0] = 1;
	for (int i = 1; i <= MAX; i++) {
		fac[i] = fac[i - 1] * i;
	}
	rfac[MAX] = fac[MAX].inverse();
	for (int i = MAX; i >= 1; i--) {
		rfac[i - 1] = rfac[i] * i;
	}
}

mint nPr(int n, int r) {
	if (!inited) init();
	return r < 0 || n < r ? 0 : fac[n] * rfac[n - r];
}

mint nCr(int n, int r) {
	if (!inited) init();
	return r < 0 || n < r ? 0 : fac[n] * rfac[n - r] * rfac[r];
}

mint nHr(int n, int r) {
	if (!inited) init();
	return r == 0 ? 1 : nCr(n + r - 1, r);
}

int main()
{
	int N, M;
	cin >> N >> M; --N;
	mint res = 0;
	for (int i = 0; i <= M / 2; i++) if (M - i >= N && (M - i) - i <= N) {
		int y = M - i - N;
		res += nCr(M, y) - nCr(M, y - 1) * 2 + nCr(M, y - 2);
	}
	cout << res << endl;
	return 0;
}
accuracy: 0.03508771929824561
time_ms: 10
memory_kb: 18696
final_rank: 5