【CF622F】The Sum of the k-th Powers (拉格朗日插值法)

用的dls的板子，因为看不懂调了好久...果然用别人的板子就是这么蛋疼- -||

num数组0~k+1储存了k+2个值，且这k+2个值是自然数i的k次方而不是次方和，dls的板子自己帮你算和的...搞得我弄了好久

 #include <iostream>
 #include <string.h>
 #include <cstdio>
 #include <vector>
 #include <queue>
 #include <assert.h>
 #include <math.h>
 #include <string>
 #include <algorithm>
 #include <functional>

 #define SIGMA_SIZE 26
 #define lson rt<<1
 #define rson rt<<1|1
 #define lowbit(x) (x&-x)
 #define foe(i, a, b) for(int i=a; i<=b; i++)
 #define fo(i, a, b) for(int i=a; i<b; i++)
 #define pii pair<int,int>
 #pragma warning ( disable : 4996 )

 using namespace std;
 typedef long long LL;
 inline LL LMax(LL a, LL b) { return a>b ? a : b; }
 inline LL LMin(LL a, LL b) { return a>b ? b : a; }
 inline LL lgcd(LL a, LL b) { return b ==  ? a : lgcd(b, a%b); }
 inline LL llcm(LL a, LL b) { return a / lgcd(a, b)*b; }  //a*b = gcd*lcm
 inline int Max(int a, int b) { return a>b ? a : b; }
 inline int Min(int a, int b) { return a>b ? b : a; }
 inline int gcd(int a, int b) { return b ==  ? a : gcd(b, a%b); }
 inline int lcm(int a, int b) { return a / gcd(a, b)*b; }  //a*b = gcd*lcm
 const LL INF = 0x3f3f3f3f3f3f3f3f;
 const LL mod = 1e9+;
 const double eps = 1e-;
 const int inf = 0x3f3f3f3f;
 const int maxk = 3e6 + ;
 const int maxn = 1e6+;

 /// 注意mod，使用前须调用一次 polysum::init(int M);
 namespace polysum {
 #define rep(i,a,n) for (int i=a;i<n;i++)
 #define per(i,a,n) for (int i=n-1;i>=a;i--)
     typedef long long ll;
     const ll mod = 1e9 + ; /// 取模值
     ll powmod(ll a, ll b) { ll res = ; a %= mod; assert(b >= ); for (; b; b >>= ) { if (b & )res = res*a%mod; a = a*a%mod; }return res; }

     const int D = ; /// 最高次限制
     ll a[D], f[D], g[D], p[D], p1[D], p2[D], b[D], h[D][], C[D];
     ll calcn(int d, ll *a, ll n) {
         if (n <= d) return a[n];
         p1[] = p2[] = ;
         rep(i, , d + ) {
             ll t = (n - i + mod) % mod;
             p1[i + ] = p1[i] * t%mod;
         }
         rep(i, , d + ) {
             ll t = (n - d + i + mod) % mod;
             p2[i + ] = p2[i] * t%mod;
         }
         ll ans = ;
         rep(i, , d + ) {
             ll t = g[i] * g[d - i] % mod*p1[i] % mod*p2[d - i] % mod*a[i] % mod;
             if ((d - i) & ) ans = (ans - t + mod) % mod;
             else ans = (ans + t) % mod;
         }
         return ans;
     }
     void init(int M) { /// M：最高次
         f[] = f[] = g[] = g[] = ;
         rep(i, , M + ) f[i] = f[i - ] * i%mod;
         g[M + ] = powmod(f[M + ], mod - );
         per(i, , M + ) g[i] = g[i + ] * (i + ) % mod;
     }
     ll polysum(ll n, ll *arr, ll m) { /// a[0].. a[m] \sum_{i=0}^{n-1} a[i]
         for (int i = ; i <= m; i++)
             a[i] = arr[i];
         a[m + ] = calcn(m, a, m + );
         rep(i, , m + ) a[i] = (a[i - ] + a[i]) % mod;
         return calcn(m + , a, n - );
     }
     ll qpolysum(ll R, ll n, ll *a, ll m) { /// a[0].. a[m] \sum_{i=0}^{n-1} a[i]*R^i
         if (R == ) return polysum(n, a, m);
         a[m + ] = calcn(m, a, m + );
         ll r = powmod(R, mod - ), p3 = , p4 = , c, ans;
         h[][] = ; h[][] = ;
         rep(i, , m + ) {
             h[i][] = (h[i - ][] + a[i - ])*r%mod;
             h[i][] = h[i - ][] * r%mod;
         }
         rep(i, , m + ) {
             ll t = g[i] * g[m +  - i] % mod;
             if (i & ) p3 = ((p3 - h[i][] * t) % mod + mod) % mod, p4 = ((p4 - h[i][] * t) % mod + mod) % mod;
             else p3 = (p3 + h[i][] * t) % mod, p4 = (p4 + h[i][] * t) % mod;
         }
         c = powmod(p4, mod - )*(mod - p3) % mod;
         rep(i, , m + ) h[i][] = (h[i][] + h[i][] * c) % mod;
         rep(i, , m + ) C[i] = h[i][];
         ans = (calcn(m, C, n)*powmod(R, n) - c) % mod;
         if (ans<) ans += mod;
         return ans;
     }
 }

 LL num[maxn];
 LL n, k;

 void init()
 {
     for( int i = ; i <= k+; i++ )
         num[i] = polysum::powmod((LL)i+, k);
 }

 int main(){
     cin >> n >> k;

     polysum::init(k);
     init();

     LL ans = polysum::polysum(n, num, k+)%mod;
     printf("%lld\n", ans);
     return ;
 }