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zc
2020-02-08 14:58:00
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E_i=\frac{F_i}{q_i}=\sum_{j=1}^{i-1}\frac{q_j}{(i-j)^2}-\sum_{j=i+1}^n \frac{q_j}{(j-i)^2}

f_i=q_i,g_i=\frac 1{i^2},f_0=0,g_0=0

E_i=\sum_{j=1}^i f_j\cdot g_{i-j}-\sum_{j=i}^n f_j\cdot g_{j-i}

可以发现前半部分是卷积的形式

我们来考虑后半部分

\sum_{j=i}^n f_j\cdot g_{j-i} \\ =\sum_{j=0}^{n-i} f_{i+j}\cdot g_j

t=n-i,f'_i=f_{n-i}

=\sum_{j=i}^t g_j\cdot f'_{t-i}

也成功转化为卷积的形式

可以开始愉快地使用FFT了

#include<bits/stdc++.h>
namespace ZDY{
    #pragma GCC optimize(3)
    #define il __inline__ __attribute__ ((always_inline))
    #define register
    #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?__=1,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=400011;
int n,limit=1,l=0,r[N];
db q[N],Pi=acos(-1.0);
struct cp{db r,i;cp(db R=0,db I=0){r=R,i=I;}}a[N],b[N],c[N];
cp operator + (cp a,cp b){return cp(a.r+b.r,a.i+b.i);}
cp operator - (cp a,cp b){return cp(a.r-b.r,a.i-b.i);}
cp operator * (cp a,cp b){return cp(a.r*b.r-a.i*b.i,a.r*b.i+a.i*b.r);}
void fft(cp *A,int typ){
    Fur(i,0,limit-1)
        if(i<r[i])SWAP(A[i],A[r[i]]);
    for(int len=1;len<limit;len<<=1){
        cp Wn(cos(Pi/len),typ*sin(Pi/len));
        for(int i=0;i<limit;i+=(len<<1)){
            cp w(1,0);
            Fur(k,0,len-1){
                cp t=w*A[i+k+len];
                A[i+k+len]=A[i+k]-t;
                A[i+k]=A[i+k]+t;
                w=w*Wn;
            }
        }
    }
    if(~typ)return;
    Fur(i,0,limit-1)A[i].r/=limit;
}
int main(){
    scanf("%d",&n);
    Fur(i,1,n)
        scanf("%lf",&a[i].r),
        b[i].r=(db)1.0/i/i,
        c[n-i]=a[i];
    while(limit<=n*2)limit<<=1,++l;
    Fur(i,0,limit-1)
        r[i]=(r[i>>1]>>1)|((i&1)<<(l-1));
    fft(a,1),fft(b,1),fft(c,1);
    Fur(i,0,limit-1)a[i]=a[i]*b[i],c[i]=c[i]*b[i];
    fft(a,-1),fft(c,-1);
    Fur(i,1,n)printf("%.3f\n",a[i].r-c[n-i].r);
}
LG 3338 [ZJOI2014]力
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