【BZOJ】【3157】&【BZOJ】【3516】国王奇遇记

数论

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　　题解：http://www.cnblogs.com/zhuohan123/p/3726933.html

　　copy一下推导过程：

令$$S_i=\sum_{k=1}^{n}k^im^k$$

我们有$$ \begin{aligned} (m-1)S_i &= mS_i-S_i \\&=\sum_{k=1}^n k^im^{k+1}-\sum_{k=1}^n k^i m^k \\&=\sum_{k=2}^{n+1}(k-1)^i m^k-\sum_{k=1}^n k^i m^k \\&=n^i m^{n+1}+\sum_{k=1}^n m^k ( (k-1)^i-k^i ) \\&=n^i m^{n+1}+\sum_{k=1}^n \big( \sum_{j=1}^{i-1}(-1)^{i-j} \binom{i}{j}k^jm^k\big) \\&=n^i m^{n+1}+\sum_{j=0}^{i-1}(-1)^{i-j}\binom{i}{j} S_j \end{aligned} $$

 /**************************************************************
     Problem: 3516
     User: Tunix
     Language: C++
     Result: Accepted
     Time:372 ms
     Memory:1300 kb
 ****************************************************************/

 //BZOJ 3157
 #include<cstdio>
 #include<cstring>
 #include<cstdlib>
 #include<iostream>
 #include<algorithm>
 #define rep(i,n) for(int i=0;i<n;++i)
 #define F(i,j,n) for(int i=j;i<=n;++i)
 #define D(i,j,n) for(int i=j;i>=n;--i)
 #define pb push_back
 using namespace std;
 typedef long long LL;
 const int N=;
 const LL P=;
 /*******************template********************/
 #define sqr(x) (x)*(x)
 LL n,m;
 LL fac[N],inv[N],s[N];
 LL C(int a,int b){return fac[a]*inv[b]%P*inv[a-b]%P;}
 LL Pow(LL a,LL b){
     LL r=;
     for(;b;b>>=,a=a*a%P) if (b&) r=r*a%P;
     return r;
 }
 int main(){
 #ifndef ONLINE_JUDGE
     freopen("3157.in","r",stdin);
     freopen("3157.out","w",stdout);
 #endif
     scanf("%lld%lld",&n,&m);
     if (m==){ printf("%lld\n",(n+)*n/%P);return ;}
     fac[]=; F(i,,m) fac[i]=fac[i-]*i%P;
     inv[m]=Pow(fac[m],P-);
     inv[]=;
     D(i,m-,) inv[i]=inv[i+]*(i+)%P;
     s[]=((Pow(m,n+)-m)%P+P)%P*Pow(m-,P-)%P;
     F(i,,m){
         s[i]=Pow(n,i)*Pow(m,n+)%P;
         rep(j,i) s[i]=((s[i]+((i-j)%== ? - : )*C(i,j)*s[j])%P+P)%P;
         s[i]=s[i]*Pow(m-,P-)%P;
     }
     printf("%lld\n",s[m]);
     return ;
 }

3157: 国王奇遇记

Time Limit: 10 Sec  Memory Limit: 512 MB
Submit: 357  Solved: 196
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Description

 

Input

共一行包括两个正整数N和M。

Output

共一行为所求表达式的值对10^9+7取模的值。

Sample Input

5 3

Sample Output

36363

HINT

1<=N<=10^9,1<=M<=200

Source

Katharon+#1

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