牛客网暑期ACM多校训练营(第一场) F Sum of Maximum

杜教板子：

证明https://blog.csdn.net/Lee_w_j__/article/details/81135539

#include <cstdio>
#include <iostream>
#include <vector>
#include <cstring>
#include <algorithm>
#include <string>
#include <map>
#include <set>
#include <list>
#include <cmath>
#include <cstring>
#include <queue>
#include <stack>
#include <ctime>
#include <complex>
using namespace std;
#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--)
#define forn(i,n) for(int i = 0;i<n;i++)
#define for1(i,n) for(int i = 1;i<=n;i++)
#define pb push_back
//#define mp make_pair
#define all(x) (x).begin(),(x).end()
//#define fi first
#define se second
#define SZ(x) ((int)(x).size())
typedef vector<int> VI;
typedef long long ll;
typedef long double ld;
typedef pair<int,int> PII;
typedef pair<ll,ll> PLL;
typedef unsigned us;
typedef unsigned long long ull;
const ll mod=1e9+7;
const int inf = 1000000000;
const int maxn = 1003005;
const int maxa = 300005;
ll gcd(ll a,ll b) { return b?gcd(b,a%b):a;}
ll powmod(ll a,ll b) {ll res=1;a%=mod; if(b<0) return -1; for(;b;b>>=1){if(b&1)res=res*a%mod;a=a*a%mod;}return res;}
int INF = 10000000;
//ll fac(ll a) {ll ans = 1;for(int i = 1;i<=a;i++) ans*=i;return a==0?1:ans;};
//start = clock();
//finish = clock();
//cout<<double(finish - start) / CLOCKS_PER_SEC<<" seconds"<<endl;
//rep(i,2,100002) inv[i] = inv[mod%i]*(mod-mod/i)%mod; //
clock_t start,finish;
//double duration;
//long double eps = 1e-10;

//You wanna hack this? Are you Serious?

namespace polysum {
    const int D=3001000;
    ll a[D],f[D],g[D],p[D],p1[D],p2[D],b[D],h[D][2],C[D];
    ll calcn(int d,ll *a,ll n) {
        if (n<=d) return a[n];
        p1[0]=p2[0]=1;
        rep(i,0,d+1) {
            ll t=(n-i+mod)%mod;
            p1[i+1]=p1[i]*t%mod;
        }
        rep(i,0,d+1) {
            ll t=(n-d+i+mod)%mod;
            p2[i+1]=p2[i]*t%mod;
        }
        ll ans=0;
        rep(i,0,d+1) {
            ll t=g[i]*g[d-i]%mod*p1[i]%mod*p2[d-i]%mod*a[i]%mod;
            if ((d-i)&1) ans=(ans-t+mod)%mod;
            else ans=(ans+t)%mod;
        }
        return ans;
    }
    void init(int M) {
        f[0]=f[1]=g[0]=g[1]=1;
        rep(i,2,M+5) f[i]=f[i-1]*i%mod;
        g[M+4]=powmod(f[M+4],mod-2);
        per(i,1,M+4) g[i]=g[i+1]*(i+1)%mod;
    }
    ll polysum(ll n,ll *a,ll m) { // a[0].. a[m] \sum_{i=0}^{n-1} a[i]
        a[m+1]=calcn(m,a,m+1);
        rep(i,1,m+2) a[i]=(a[i-1]+a[i])%mod;
        return calcn(m+1,a,n-1);
    }
    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==1) return polysum(n,a,m);
        a[m+1]=calcn(m,a,m+1);
        ll r=powmod(R,mod-2),p3=0,p4=0,c,ans;
        h[0][0]=0;h[0][1]=1;
        rep(i,1,m+2) {
            h[i][0]=(h[i-1][0]+a[i-1])*r%mod;
            h[i][1]=h[i-1][1]*r%mod;
        }
        rep(i,0,m+2) {
            ll t=g[i]*g[m+1-i]%mod;
            if (i&1) p3=((p3-h[i][0]*t)%mod+mod)%mod,p4=((p4-h[i][1]*t)%mod+mod)%mod;
            else p3=(p3+h[i][0]*t)%mod,p4=(p4+h[i][1]*t)%mod;
        }
        c=powmod(p4,mod-2)*(mod-p3)%mod;
        rep(i,0,m+2) h[i][0]=(h[i][0]+h[i][1]*c)%mod;
        rep(i,0,m+2) C[i]=h[i][0];
        ans=(calcn(m,C,n)*powmod(R,n)-c)%mod;
        if (ans<0) ans+=mod;
        return ans;
    }
}
ll n,k,R;
ll a[maxn];
int main() {
    ios::sync_with_stdio(0);
    cin>>n>>R>>k;
    for(int i = 0;i<=2010;i++) a[i] = powmod(i,k);
    polysum::init(k+5);
    ll ans = 0;
    ans = polysum::qpolysum(R,n+1,a,k+1);
    if(k==0) ans = (ans-1+mod)%mod;
    cout<<ans<<endl;





}

不是我说dls也太强了吧，，随便一个namespace全世界都在用。。。