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\sum_{i=0}^k {n \choose i} \bmod 2333

p=2333 \sum_{i=0}^k {n \choose i} \bmod p \\ =\sum_{i=0}^k {\left \lfloor \frac np \right \rfloor \choose \left \lfloor \frac ip \right \rfloor} {n \bmod p \choose i\bmod p}\bmod p \\ ={\left \lfloor \frac np \right \rfloor \choose 0}\sum_{i=0}^{p-1}{n \bmod p \choose i}+{\left \lfloor \frac np \right \rfloor \choose 1}\sum_{i=0}^{p-1}{n \bmod p \choose i}+ \cdots+ {\left \lfloor \frac np \right \rfloor \choose \left \lfloor \frac kp \right \rfloor}\sum_{i=0}^{k \bmod p}{n \bmod p \choose i}

f(n,k)=\sum_{i=0}^k {n \choose i}

原式可以转化为:

f(n \bmod p,p-1) \cdot f(\left \lfloor \frac np \right \rfloor,\left \lfloor \frac kp \right \rfloor -1)+{\left \lfloor \frac np \right \rfloor \choose \left \lfloor \frac kp \right \rfloor} f(n\bmod p,k\bmod p)

#include<bits/stdc++.h>
namespace ZDY{
    #pragma GCC optimize(3)
    #define il __inline__ __attribute__ ((always_inline))
    #define ll long long
    #define ull unsigned long long
    #define db double
    #define sht short
    #define MB template <class T>il
    #define Fur(i,x,y) for(int i(x);i<=y;++i)
    #define Fdr(i,x,y) for(int i(x);i>=y;--i)
    #define fl(i,x) for(int i(head[x]),to;to=e[i].to,i;i=e[i].nxt)
    #define clr(x,y) memset(x,y,sizeof(x))
    #define cpy(x,y) memcpy(x,y,sizeof(x))
    #define fin(s) freopen(s".in","r",stdin)
    #define fout(s) freopen(s".out","w",stdout)
    #define fcin ios::sync_with_stdio(false)
    #define l2(n) ((int)(log2(n)))
    #define inf 2122219134
    MB T ABS(T x){return x>0?x:-x;}
    MB T MAX(T x,T y){return x>y?x:y;}
    MB T MIN(T x,T y){return x<y?x:y;}
    MB T GCD(T x,T y){return y?GCD(y,x%y):x;}
    MB void SWAP(T &x,T &y){T t=x;x=y;y=t;}
}using namespace ZDY;using namespace std;
namespace IO{const int str=1<<20;static char in_buf[str],*in_s,*in_t;bool __=0;il char gc(){return (in_s==in_t)&&(in_t=(in_s=in_buf)+fread(in_buf,1,str,stdin)),in_s==in_t?EOF:*in_s++;}il void in(string &ch){ch.clear();if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF){__=1;return;}ch+=c;while((c=gc())!=EOF&&!isspace(c))ch+=c;if(c==EOF)__=1;}il void in(char &ch){if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF)__=1;else ch=c;}il void in(char *ch){*ch='\0';if(__)return;char c;while((c=gc())!=EOF&&isspace(c));if(c==EOF){__=1;return;}*ch=c;ch++;while((c=gc())!=EOF&&!isspace(c))*ch=c,ch++;if(c==EOF)__=1;*ch='\0';}template<typename T>il void in(T &x){if(__)return;char c=gc();bool f=0;while(c!=EOF&&(c<'0'||c>'9'))f^=(c=='-'),c=gc();if(c==EOF){__=1;return;}x=0;while(c!=EOF&&'0'<=c&&c<='9')x=x*10+c-48,c=gc();if(c==EOF)__=1;if(f)x=-x;}template<typename T,typename ... arr>il void in(T &x,arr & ... y){in(x),in(y...);}const char ln='\n';static char out_buf[str],*out_s=out_buf,*out_t=out_buf+str;il void flush(){fwrite(out_buf,1,out_s-out_buf,stdout);out_s=out_buf;}il void pt(char c){(out_s==out_t)?(fwrite(out_s=out_buf,1,str,stdout),*out_s++=c):(*out_s++=c);}il void out(const char* s){while(*s)pt(*s++);}il void out(char* s){while(*s)pt(*s++);}il void out(char c){pt(c);}il void out(string s){for(int i=0;s[i];i++)pt(s[i]);}template<typename T>il void out(T x){if(!x){pt('0');return;}if(x<0)pt('-'),x=-x;char a[50],t=0;while(x)a[t++]=x%10,x/= 10;while(t--)pt(a[t]+'0');}template<typename T,typename ... arr>il void out(T x,arr & ... y){out(x),out(y...);}}using namespace IO;
const int N=2335,p=2333;
ll C[N+11][N+11],F[N+11][N+11];
void init(){
    C[0][0]=1;
    Fur(i,1,N)C[i][0]=C[i][i]=1;
    Fur(i,1,N)Fur(j,1,i-1)C[i][j]=(C[i-1][j]+C[i-1][j-1])%p;
    F[0][0]=1;
    Fur(i,1,N)F[i][0]=1;
    Fur(i,0,N)Fur(j,1,N)F[i][j]=(C[i][j]+F[i][j-1])%p;
}
ll lucas(ll n,ll m){
    if(n<m)return 0;
    if(!n)return 1;
    return lucas(n/p,m/p)*C[n%p][m%p]%p;
}
ll f(ll n,ll k){
    if(k<0)return 0;
    if(!n||!k)return 1;
    if(n<p&&k<p)return F[n][k];
    return (f(n/p,k/p-1)*F[n%p][p-1]%p+lucas(n/p,k/p)*f(n%p,k%p)%p)%p;
}
int main(){
    int T;ll n,k;
    in(T);init();
    while(T--){
        in(n,k);
        out(f(n,k),ln);
    }
    flush();
}
LG 4345 [SHOI2015]超能粒子炮·改
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