题目链接:http://acm.hdu.edu.cn/showproblem.php?pid=3483
A Very Simple Problem
Time Limit: 4000/2000 MS (Java/Others) Memory Limit: 65536/32768 K (Java/Others)
Total Submission(s): 945 Accepted Submission(s): 471
Problem Description
This is a very simple problem. Given three integers N, x, and M, your task is to calculate out the following value:
Input
There are several test cases. For each case, there is a line with three integers N, x, and M, where 1 ≤ N, M ≤ 2*109, and 1 ≤ x ≤ 50.
The input ends up with three negative numbers, which should not be processed as a case.
Output
For each test case, print a line with an integer indicating the result.
Sample Input
100 1 10000
3 4 1000
-1 -1 -1
Sample Output
5050
444
Source
2010 ACM-ICPC Multi-University Training Contest(5)——Host by BJTU
1 //计算排列数(杨辉三角) 2 //C(m,n) = C(m-1,n-1)+C(m-1,n) 3 //快速幂 4 /* 5 * [题意] 6 * 输入n, x, m 7 * 求(1^x)*(x^1)+(2^x)*(x^2)+(3^x)*(x^3)+...+(n^x)*(x^n) 8 * [解题方法] 9 * 设f[n] = [x^n, n*(x^n), (n^2)*(x^n),..., (n^x)*(x^n)] 10 * 则f[n][k] = (n^k)*(x^n) 11 * 问题转化为求:( g[n] = f[1][x]+f[2][x]+...+f[n][x] ) 12 * 设C(i,j)为组合数,即i种元素取j种的方法数 13 * 所以有:f[n+1][k] = ((n+1)^k)*(x^(n+1)) (二次多项式展开) 14 * = x*( C(k,0)*(x^n) +C(k,1)*n*(x^n)+...+C(k,k)*(n^k)*(x^n) ) 15 * = x*( C(k,0)*f[n][0]+C(k,1)*f[n][1]+...+C(k,k)*f[n][k] ) 16 * 所以得: 17 * |x*1 0................................0| |f[n][0]| |f[n+1][0]| 18 * |x*1 x*1 0............................0| |f[n][1]| |f[n+1][1]| 19 * |x*1 x*2 x*1 0........................0| * |f[n][2]| = |f[n+1][2]| 20 * |......................................| |.......| |.........| 21 * |x*1 x*C(k,1) x*C(k,2)...x*C(k,x) 0...0| |f[n][k]| |f[n+1][k]| 22 * |......................................| |.......| |.........| 23 * |x*1 x*C(x,1) x*C(x,2).......x*C(x,x) 0| |f[n][x]| |f[n+1][x]| 24 * |0................................0 1 1| |g[n-1] | | g[ n ] | 25 */ 26 #include<cstdio> 27 #include<cstring> 28 #include<cmath> 29 #include<algorithm> 30 using namespace std; 31 #define ll long long 32 const ll maxn = 55; 33 ll c[maxn][maxn]; 34 ll n, mod, x, m; 35 struct Mat{ 36 ll f[maxn][maxn]; 37 }; 38 void init() 39 { 40 ll i,j,k; 41 c[0][0] = c[1][0] = c[1][1] = 1; 42 for(i = 2; i < maxn; i++){ 43 c[i][0] = c[i][i] = 1; 44 for(j = 1; j < i; j++){ 45 c[i][j] = c[i-1][j]+c[i-1][j-1]; 46 } 47 } 48 } 49 Mat operator *(Mat a, Mat b) 50 { 51 ll i, j, k; 52 Mat c; 53 memset(c.f,0,sizeof(c.f)); 54 for(k = 0; k < m; k++){ 55 for(i = 0; i < m; i++){ 56 for(j = 0; j < m; j++){ 57 if(!b.f[k][j]) continue; 58 c.f[i][j] = (c.f[i][j]+(a.f[i][k]*b.f[k][j])%mod)%mod; 59 } 60 } 61 } 62 return c; 63 } 64 Mat multi(Mat a,ll b) 65 { 66 Mat s; 67 memset(s.f,0,sizeof(s.f)); 68 for(int i = 0; i < m; i++){ 69 s.f[i][i] = 1; 70 } 71 while(b){ 72 if(b&1) s = s*a; 73 a = a*a; 74 b>>=1; 75 } 76 return s; 77 } 78 int main() 79 { 80 init(); 81 while(~scanf("%lld%lld%lld",&n,&x,&mod)) 82 { 83 if(n<0&&x<0&&mod<0) break; 84 Mat e; 85 ll i, j; 86 ll ans = 0; 87 memset(e.f,0,sizeof(e.f)); 88 for(i = 0; i <= x; i++){ 89 for(j = i; j <= x; j++){ 90 e.f[j][i] = c[x-i][j-i]*x%mod; 91 } 92 } 93 e.f[0][x+1] = e.f[x+1][x+1] = 1; 94 m = x+2; 95 e = multi(e,n); 96 for(i = 0; i < m-1; i++) ans = (ans+x*e.f[i][m-1])%mod; 97 printf("%lld\n",(ans+mod)%mod); 98 } 99 return 0; 100 }