令a[i]=q[i],b[i]=1/(i^2),则E[j]=∑(1≤j≤i-1)a[j]*b[i-j]-E[j]=∑(i+1≤j≤n)a[j]*b[j-i]。
左边就是卷积形式,直接FFT;右边和bzoj2194等价,翻转一个数组后FFT。
#include<iostream> #include<cstdio> #include<cmath> #include<cstdlib> #include<cstring> #include<algorithm> #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 ll long long #define maxn 500005 using namespace std; int nn,n,m,len,rev[maxn]; double q[maxn],ans[maxn]; const double pi=acos(-1.0); struct cp { double x,y; cp(double xx=0,double yy=0){x=xx;y=yy;} inline cp operator +(cp b){return (cp){x+b.x,y+b.y};} inline cp operator -(cp b){return (cp){x-b.x,y-b.y};} inline cp operator *(cp b){return (cp){x*b.x-y*b.y,x*b.y+y*b.x};} }a[maxn],b[maxn],c[maxn]; inline void fft(cp *x,int n,int flag) { F(i,0,n-1) if (rev[i]>i) swap(x[i],x[rev[i]]); for(int m=2;m<=n;m<<=1) { cp wn=cp(cos(2.0*pi/m*flag),sin(2.0*pi/m*flag)); for(int i=0;i<n;i+=m) { cp w=cp(1.0); int mid=m>>1; F(j,0,mid-1) { cp u=x[i+j],v=x[i+j+mid]*w; x[i+j]=u+v;x[i+j+mid]=u-v; w=w*wn; } } } if (flag==-1) F(i,0,n-1) x[i].x/=n; } int main() { scanf("%d",&n);nn=n; F(i,0,n-1) scanf("%lf",&q[i]); n=n*2-1;m=1; while (m<n) m<<=1,len++; n=m; F(i,0,n-1) { int x=i,y=0; F(j,1,len) (y<<=1)|=(x&1),x>>=1; rev[i]=y; } F(i,0,nn-1) a[i]=cp(q[i],0); F(i,nn,n-1) a[i]=cp(); b[0]=cp(); F(i,1,nn-1) b[i]=cp(1.0/((double)i*(double)i)); F(i,nn,n-1) b[i]=cp(); fft(a,n,1);fft(b,n,1); F(i,0,n-1) c[i]=a[i]*b[i]; fft(c,n,-1); F(i,0,nn-1) ans[i]=c[i].x; F(i,0,nn-1) a[i]=cp(q[nn-1-i],0); F(i,nn,n-1) a[i]=cp(); b[0]=cp(); F(i,1,nn-1) b[i]=cp(1.0/((double)i*(double)i)); F(i,nn,n-1) b[i]=cp(); fft(a,n,1);fft(b,n,1); F(i,0,n-1) c[i]=a[i]*b[i]; fft(c,n,-1); F(i,0,nn-1) ans[i]-=c[nn-1-i].x; F(i,0,nn-1) printf("%.5lf\n",ans[i]); return 0; }