【紀中集訓2019.3.12】Z的禮物

題意

已知\(a_{i} = \sum_{j=1}^{i} \{^{i} _{j} \}b_{j}\), 給出\(a_{1} 到 a_{n}\) ；

求\(b_{l} 到 b_{r}\)在\(1e9+7\)的意義下取模的值；

\(1 \le l \le r \le n \le 10^5\)

\(r-l \le 100\)

\(0 \le a_{i} \lt 10^9 + 7\)

題解

• Part1

• 斯特林反演（http://www.javashuo.com/article/p-qionilex-kr.html）：

  • \(f(n) = \sum_{i=1}^{n} \{^n_i\} g(n) \Longleftrightarrow g(n) = \sum_{i=1}^{n} (-1)^{n-i} [^n_i]f(i)\)

• 因爲\(r-l\)比較小，能夠直接暴力求，只須要快速求出\([^n_1] - [^n_n]\)

• \([_m^n] = [x^m] x^{\overline n}\) , 其中\(x^{\overline n} = \Pi_{i=1}^n(x+i-1)\)

• Part 2

• 直接作是\(n log ^2n\)的，假設在計算\(F_n(x) = x^{\overline n}\)時已經計算了\(F_\frac{n}{2}(x)\),記\(m = \frac{n}{2} , F_{m} = \sum_{i=0}^{m}a_{i} x^{i}\)

• 只須要計算\(F'_{m} (x) = F_{m}(x+m)\)和\(F_{m}\)相乘；

• \[ F'_m(x) = \sum_{i=0}^{m} a_{i} (x+m)^i \\ = \sum_{i=0}^{m} a _{i}\sum_{j=0}^{i}(^i_j)x^j m^{i-j} \\ = \sum_{i=0}^{m} \sum_{j=0}^{i} a_{i} \frac{i!}{j!(i-j)!}x^{j}m^{i-j} \\ = \sum_{i=0}^{m} \frac{x^i}{i!} \sum_{j=0}^{m-i} a_{i+j}(i+j)! \ \frac{m^{j}}{j!} \]

• 記$A(x)=\sum_{i=0}^{m} a_{i} i! x^{i}, B(x) = \sum_{i=0}^{m}\frac{m^{i}}{i!} x^{m-i} $

• 相乘以後取第\(m+i\)項能夠算後面的東西

• Part 3

• 但是不是\(ntt\)模數，直接取模會爆\(long \ long\)須要用到拆係數\(ntt\);

• 將\(A(x)*B(x)\)分解成\((k * A1(x) + A2(x)) * (k * B1(x) + B2(x)) , k 通常取\sqrt{n} (2^{15} ) 左右\)

• 暴力是7次\(fft\)，能夠優化到須要4次\(fft\)

• 兩次\(fft\)合併成一次的作法，須要計算\(dft(A(x))和dft(B(x))\)

• 記： \[\begin{align} P(x) = A(x) + iB(x)\\ Q(x) = A(x) -iB(x) \end{align}\]

• 考慮直接作\(P\)爲\(P’\),考慮\(Q \ dft\)以後的結果\(Q'\)

  \[ \begin{align} Q'[k] &= A(w_{n}^{k}) - iB(w_{n}^{k}) \\ &= \sum_{j=0}^{n-1} (a_{j}-ib_{j})w^{jk}_{n} \\ &= \sum_{j=0}^{n-1}(a_{j}-ib_{j})(cos(\frac{2\pi jk}{n} ) + isin(\frac{2\pi jk}{n}) ) \\ &= \sum_{j=0}^{n-1}(a_{j}cos(\frac{2\pi jk}{n}) + b_{j}sin(\frac{2\pi jk}{n}))- i(b_{j}cos(\frac{2\pi jk}{n}) - a_{j}sin(\frac{2\pi jk}{n})) \\ &= conj \sum_{j=0}^{n-1} (a_{j}cos(\frac{-2\pi jk}{n}) - b_{j}sin(\frac{-2\pi jk}{n} ) ) +i( b_{j}cos(\frac{-2\pi jk}{n}) + a_{j}sin(\frac{-2\pi jk}{n})) \\ &= conj \sum_{j=0}^{n-1} (a_{j}+ib_{j})(cos(\frac{-2\pi jk}{n})+isin(\frac{-2\pi jk}{n})) \\ &= conj \ \sum_{j=0}^{n-1} (a_{j}+ib_{j})w^{-jk}_{n} \\ &= conj \ P'[n-k] \end{align} \]

• 因此由\(P'\)直接獲得\(Q'\):

• 即：\[ \begin{align} A'(x) = \frac{P'(x) + Q'(x)}{2} \\ B'(x) = \frac{P'(x) - Q'(x)}{2i}\end{align} g \]

• 結果也存在實部和虛部裏直接\(idft\)回去就行了；

• 注意精度；

  #include<bits/stdc++.h>
  #define ld long double
  #define ll long long 
  using namespace std;
  const int N=400010,mod=1e9+7;
  const ld pi=acos(-1);
  int n,l,r,rev[N],fac[N],inv[N],p[N],a[N],b[N];
  char gc(){
    static char*p1,*p2,s[1000000];
    if(p1==p2)p2=(p1=s)+fread(s,1,1000000,stdin);
    return(p1==p2)?EOF:*p1++;
  }
  int rd(){
    int x=0;char c=gc();
    while(c<'0'||c>'9')c=gc();
    while(c>='0'&&c<='9')x=(x<<1)+(x<<3)+c-'0',c=gc();
    return x;
  }
  int pw(int x,int y){
    int re=1;
    while(y){
        if(y&1)re=(ll)re*x%mod;
        y>>=1,x=(ll)x*x%mod;
    }
    return re;
  }
  struct C{
    ld x,y;
    C(ld _x=0,ld _y=0):x(_x),y(_y){};
    C operator +(const C&A)const{return C(x+A.x,y+A.y);}
    C operator -(const C&A)const{return C(x-A.x,y-A.y);}
    C operator *(const C&A)const{return C(x*A.x-y*A.y,x*A.y+y*A.x);}
    C operator /(const ld&A)const{return C(x/A,y/A);}
    C operator !()const{return C(x,-y);}
  };
  void fft(C*A,int len,int f){
    for(int i=0;i<len;++i)if(i<rev[i])swap(A[i],A[rev[i]]);
    for(int i=1;i<len;i<<=1){
        C wn=C(cos(pi/i),f*sin(pi/i));
        for(int j=0;j<len;j+=i<<1){
            C w=C(1,0);
            for(int k=0;k<i;++k,w=w*wn){
                C x=A[j+k],y=w*A[j+k+i];
                A[j+k]=x+y,A[j+k+i]=x-y;
            }
        }
    }
    if(!~f){for(int i=0;i<len;++i){A[i]=A[i]/len;}}
  }
  void mul(int*A,int*B,int n){
    static C t1[N],t2[N],t3[N],t4[N];
    int L,len;
    for(L=0,len=1;len<(n+1)<<1;++L,len<<=1);
    for(int i=0;i<len;++i){rev[i]=(rev[i>>1]>>1)|((i&1)<<(L-1));}
    for(int i=0;i<=n;++i)t1[i]=C(A[i]>>15,A[i]&0x7fff),t2[i]=C(B[i]>>15,B[i]&0x7fff);
    for(int i=n+1;i<len;++i)t1[i]=t2[i]=C(0,0);
    fft(t1,len,1),fft(t2,len,1);
    for(int i=0;i<len;++i){
        C x1=t1[i],y1=!t1[(len-i)&(len-1)];
        C x2=t2[i],y2=!t2[(len-i)&(len-1)];
        t3[i]=(x1+y1)*x2*C(0.5,0);
        t4[i]=(x1-y1)*x2*C(0,-0.5);
    //  t3[i]=((x1+y1)*(x2+y2)+(x1+y1)*(x2-y2))*C(0.25,0);
    //  t4[i]=((x1-y1)*(x2+y2)+(x1-y1)*(x2-y2))*C(0,-0.25);
    }
    fft(t3,len,-1),fft(t4,len,-1);
    for(int i=0;i<=n<<1;++i){
        A[i]=((((ll)(t3[i].x+0.1)%mod)<<30)+(((ll)(t3[i].y+t4[i].x+0.1)%mod)<<15)+(ll)(t4[i].y+0.1))%mod;
    }
  }
  void solve(int*A,int n){
    if(n==0){A[0]=1;return;}
    if(n==1){A[1]=1;return;}
    if(n&1){
        solve(A,n-1);
        for(int i=n;i;--i)A[i]=(A[i-1]+(ll)A[i]*(n-1))%mod;
        A[0]=0; 
        return;
    }
    static int t1[N],t2[N];
    int m=n>>1;
    solve(A,m);
    for(int i=0;i<=m;++i)t1[i]=(ll)A[i]*fac[i]%mod;
    for(int i=0,t=1;i<=m;++i,t=(ll)t*m%mod)t2[m-i]=(ll)t*inv[i]%mod;
    mul(t1,t2,m);
    for(int i=0;i<=m;++i){t1[i]=(ll)inv[i]*t1[m+i]%mod;}
    mul(A,t1,m);
  }
  int main(){
    freopen("gift.in","r",stdin);
    freopen("gift.out","w",stdout);
    n=rd();l=rd();r=rd();
    for(int i=1;i<=n;++i)p[i]=rd();
    for(int i=fac[0]=inv[0]=1;i<=n;++i){
        fac[i]=(ll)fac[i-1]*i%mod;
        inv[i]=pw(fac[i],mod-2);
    }
    solve(a,l-1);
    for(int i=l-1;i<=r;++i){
        for(int j=1;j<=i;++j){
            int t=(ll)a[j]*p[j]%mod;
            if((i-j)&1)b[i]=(b[i]-t+mod)%mod; 
            else b[i]=(b[i]+t)%mod;
        }
        for(int j=i+1;j;--j){a[j]=(a[j-1]+(ll)a[j]*i)%mod;}a[0]=0;
    //  b[i]=(b[i]-b[i-1]+mod)%mod;
    }
    for(int i=l;i<=r;++i)printf("%d ",(b[i]-b[i-1]+mod)%mod);
    return 0;
  }