二项式定理+前缀Sigma

https://hihocoder.com/problemset/problem/1430

思路：

要用前缀去推Sigma总公式，比较方便。https://blog.csdn.net/weixin_37517391/article/details/83821752

 #define IOS ios_base::sync_with_stdio(0); cin.tie(0);
 #include <cstdio>//sprintf islower isupper
 #include <cstdlib>//malloc  exit strcat itoa system("cls")
 #include <iostream>//pair
 #include <fstream>//freopen("C:\\Users\\13606\\Desktop\\草稿.txt","r",stdin);
 #include <bitset>
 //#include <map>
 //#include<unordered_map>
 #include <vector>
 #include <stack>
 #include <set>
 #include <string.h>//strstr substr
 #include <string>
 #include <time.h>//and(((unsigned)time(NULL))); Seed n=rand()%10 - 0~9;
 #include <cmath>
 #include <deque>
 #include <queue>//priority_queue<int, vector<int>, greater<int> > q;//less
 #include <vector>//emplace_back
 //#include <math.h>
 //#include <windows.h>//reverse(a,a+len);// ~ ! ~ ! floor
 #include <algorithm>//sort + unique : sz=unique(b+1,b+n+1)-(b+1);+nth_element(first, nth, last, compare)
 using namespace std;//next_permutation(a+1,a+1+n);//prev_permutation
 #define rint register int
 #define fo(a,b,c) for(rint a=b;a<=c;++a)
 #define fr(a,b,c) for(rint a=b;a>=c;--a)
 #define mem(a,b) memset(a,b,sizeof(a))
 #define pr printf
 #define sc scanf
 #define ls rt<<1
 #define rs rt<<1|1
 typedef long long ll;
 void swapp(int &a,int &b);
 double fabss(double a);
 int maxx(int a,int b);
 int minn(int a,int b);
 int Del_bit_1(int n);
 int lowbit(int n);
 int abss(int a);
 const double E=2.718281828;
 const double PI=acos(-1.0);
 //const ll INF=(1LL<<60);
 const int inf=(<<);
 const double ESP=1e-;
 const int mod=(int)1e9+;
 const int N=(int)5e4+;

 long long C[][];
 void Init() {
     C[][] = ;
     for(int i = ;i <= ;++i) {
         C[i][] = ;
         for(int j = ;j <= i;++j) {
             C[i][j] = (C[i-][j-] + C[i-][j]) % mod;
         }
     }
 }
 //======================================================Cnk预处理;
 ll S[][N],SS[][N],ans[N];
 char s[N];

 void solve()
 {
     ll n,k;
     sc("%lld%lld",&n,&k);
     sc("%s",s);

     for(int i = ;i <= n;++i) S[][i] = ;
     for(int i = ;i <= n;++i) S[][i] = (s[i-]-'') + S[][i-] ;
     for(int i = ;i <= k;++i)
         for(int j = ;j <= n;++j)
             S[i][j] = S[][j] * S[i-][j] % mod;

     SS[][] = ;

     for(int i = ;i <= k;++i) {
         for(int j = ;j <= n;++j)
             SS[i][j] = (SS[i][j-] + S[i][j])%mod;
     }

     for(int i=;i<=n;++i)
     {
         ll sum=;
         for(int j=;j<=k;++j)
         {
             ll temp=C[k][j]*S[k-j][i]%mod*SS[j][i-]%mod;
             if(j&)
                 sum=(sum-temp+mod)%mod;
             else
                 sum=(sum+temp)%mod;
         }
         ans[i]=sum;
     }
     for(int i=;i<=n;++i)
         pr("%lld%c",ans[i],i==n?'\n':' ');
 }

 int main()
 {
     Init();
     int T;
     sc("%d",&T);
     while(T--)solve();
     return ;
 }

 /**************************************************************************************/

 int maxx(int a,int b)
 {
     return a>b?a:b;
 }

 void swapp(int &a,int &b)
 {
     a^=b^=a^=b;
 }

 int lowbit(int n)
 {
     return n&(-n);
 }

 int Del_bit_1(int n)
 {
     return n&(n-);
 }

 int abss(int a)
 {
     return a>?a:-a;
 }

 double fabss(double a)
 {
     return a>?a:-a;
 }

 int minn(int a,int b)
 {
     return a<b?a:b;
 }