Competitive_Programming/Math/lagrange.cpp at master · jonh14lk/Competitive_Programming · GitHub

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#include <bits/stdc++.h>
#include <ext/pb_ds/assoc_container.hpp>
#include <ext/pb_ds/tree_policy.hpp>
using namespace std;
using namespace __gnu_pbds;

template <class T>
using ordered_set = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;

#define int long long int
#define endl '\n'
#define pb push_back
#define pi pair<int, int>
#define pii pair<int, pi>
#define fir first
#define sec second
#define MAXN 600005
#define mod 1000000007

struct modint
{
  int val;
  modint(int v = 0) { val = v % mod; }
  int pow(int y)
  {
    modint x = val;
    modint z = 1;
    while (y)
    {
      if (y & 1)
        z *= x;
      x *= x;
      y >>= 1;
    }
    return z.val;
  }
  int inv() { return pow(mod - 2); }
  void operator=(int o) { val = o % mod; }
  void operator=(modint o) { val = o.val % mod; }
  void operator+=(modint o) { *this = *this + o; }
  void operator-=(modint o) { *this = *this - o; }
  void operator*=(modint o) { *this = *this * o; }
  void operator/=(modint o) { *this = *this / o; }
  bool operator==(modint o) { return val == o.val; }
  bool operator!=(modint o) { return val != o.val; }
  int operator*(modint o) { return ((val * o.val) % mod); }
  int operator/(modint o) { return (val * o.inv()) % mod; }
  int operator+(modint o) { return (val + o.val) % mod; }
  int operator-(modint o) { return (val - o.val + mod) % mod; }
};
struct lagrange
{
  int n;
  vector<modint> den;
  vector<modint> y;
  vector<modint> fat;
  vector<modint> inv_fat;

  lagrange(vector<modint> &v) // f(i) = v[i], gera um polinomio de grau n - 1
  {
    n = v.size();
    calc(n);
    calc_den(n);
    y = v;
  }
  void calc_den(int n)
  {
    den.resize(n);
    for (int i = 0; i < n; i++)
    {
      den[i] = inv_fat[n - i - 1] * inv_fat[i];
      if ((n - i - 1) % 2 == 1)
      {
        int x = (mod - den[i].val) % mod;
        den[i] = x;
      }
    }
  }
  void calc(int n)
  {
    fat.resize(n + 1);
    inv_fat.resize(n + 1);
    fat[0] = 1;
    inv_fat[0] = 1;
    for (int i = 1; i <= n; i++)
    {
      fat[i] = fat[i - 1] * i;
      inv_fat[i] = fat[i].inv();
    }
  }
  modint get_val(int x) // complexidade: O(n)
  {
    x %= mod;
    vector<modint> l(n);
    vector<modint> r(n);
    l[0] = 1, r[n - 1] = 1;
    for (int i = 1; i < n; i++)
    {
      modint cof = (x - (i - 1) + mod);
      l[i] = l[i - 1] * cof;
    }
    for (int i = n - 2; i >= 0; i--)
    {
      modint cof = (x - (i + 1) + mod);
      r[i] = r[i + 1] * cof;
    }
    modint ans = 0;
    for (int i = 0; i < n; i++)
    {
      modint cof = l[i] * r[i];
      ans += modint(cof * y[i]) * den[i];
    }
    return ans;
  }
  vector<modint> find_coefs() // encontra os coeficientes do polinomio
  {
    int nn = n;
    int d = nn - 1;
    vector<modint> c(nn, 0);
    for (int i = 0; i < y.size(); i++)
    {
      c[d] += (y[i] * den[i]);
    }
    for (int p = nn - 2; p >= 0; p--)
    {
      nn--;
      calc_den(nn);
      for (int i = 0; i <= p; i++)
      {
        y[i] -= (c[p + 1] * modint(i).pow(d));
        c[p] += (y[i] * den[i]);
      }
      d--;
    }
    return c;
  }
};
signed main()
{
  ios_base::sync_with_stdio(false);
  cin.tie(NULL);
  int n, k;
  cin >> n >> k;
  vector<modint> v;
  v.pb(0);
  int lim = k + 1;
  for (int i = 1; i <= lim; i++)
    v.pb(v.back() + modint(i).pow(k));
  lagrange l(v);
  cout << l.get_val(n).val << endl;
  return 0;
}
// https://codeforces.com/contest/622/problem/F
// https://codeforces.com/contest/1817/problem/C
// https://codeforces.com/gym/103388/problem/A