拉格朗日插值:
/// 注意mod,使用前须调用一次 polysum::init(int M); namespace polysum { #define rep(i,a,n) for (int i=a;i<n;i++) #define per(i,a,n) for (int i=n-1;i>=a;i--) typedef long long ll; const ll mod=1e9+7; /// 取模值 ll powmod(ll a,ll b) {ll res=1;a%=mod; assert(b>=0); for(;b;b>>=1){if(b&1)res=res*a%mod;a=a*a%mod;}return res;} const int D=101000; /// 最高次限制 ll a[D],f[D],g[D],p[D],p1[D],p2[D],b[D],h[D][2],C[D]; ll calcn(int d,ll *a,ll n) { if (n<=d) return a[n]; p1[0]=p2[0]=1; rep(i,0,d+1) { ll t=(n-i+mod)%mod; p1[i+1]=p1[i]*t%mod; } rep(i,0,d+1) { ll t=(n-d+i+mod)%mod; p2[i+1]=p2[i]*t%mod; } ll ans=0; rep(i,0,d+1) { ll t=g[i]*g[d-i]%mod*p1[i]%mod*p2[d-i]%mod*a[i]%mod; if ((d-i)&1) ans=(ans-t+mod)%mod; else ans=(ans+t)%mod; } return ans; } void init(int M) { /// M:最高次 f[0]=f[1]=g[0]=g[1]=1; rep(i,2,M+5) f[i]=f[i-1]*i%mod; g[M+4]=powmod(f[M+4],mod-2); per(i,1,M+4) g[i]=g[i+1]*(i+1)%mod; } ll polysum(ll n,ll *arr,ll m) { // a[0].. a[m] \sum_{i=0}^{n} a[i] for(int i = 0; i <= m; i++) a[i] = arr[i]; a[m+1]=calcn(m,a,m+1); rep(i,1,m+2) a[i]=(a[i-1]+a[i])%mod; return calcn(m+1,a,n); } ll qpolysum(ll R,ll n,ll *a,ll m) { // a[0].. a[m] \sum_{i=0}^{n-1} a[i]*R^i if (R==1) return polysum(n,a,m); a[m+1]=calcn(m,a,m+1); ll r=powmod(R,mod-2),p3=0,p4=0,c,ans; h[0][0]=0;h[0][1]=1; rep(i,1,m+2) { h[i][0]=(h[i-1][0]+a[i-1])*r%mod; h[i][1]=h[i-1][1]*r%mod; } rep(i,0,m+2) { ll t=g[i]*g[m+1-i]%mod; if (i&1) p3=((p3-h[i][0]*t)%mod+mod)%mod,p4=((p4-h[i][1]*t)%mod+mod)%mod; else p3=(p3+h[i][0]*t)%mod,p4=(p4+h[i][1]*t)%mod; } c=powmod(p4,mod-2)*(mod-p3)%mod; rep(i,0,m+2) h[i][0]=(h[i][0]+h[i][1]*c)%mod; rep(i,0,m+2) C[i]=h[i][0]; ans=(calcn(m,C,n)*powmod(R,n)-c)%mod; if (ans<0) ans+=mod; return ans; } }
例题:https://ac.nowcoder.com/acm/contest/139/F?&headNav=www
代码:
#pragma GCC optimize(2) #pragma GCC optimize(3) #pragma GCC optimize(4) #include<bits/stdc++.h> using namespace std; #define y1 y11 #define fi first #define se second #define pi acos(-1.0) #define LL long long #define ls rt<<1, l, m #define rs rt<<1|1, m+1, r //#define mp make_pair #define pb push_back #define ULL unsigned LL #define pll pair<LL, LL> #define pli pair<LL, int> #define pii pair<int, int> #define piii pair<pii, int> #define pdi pair<double, int> #define pdd pair<double, double> #define mem(a, b) memset(a, b, sizeof(a)) #define debug(x) cerr << #x << " = " << x << "\n"; #define fio ios::sync_with_stdio(false);cin.tie(0);cout.tie(0); //head const int N = 1e3 + 5; const int MOD = 1e9 + 7; int a[N], n; LL b[N]; /// 注意mod,使用前须调用一次 polysum::init(int M); namespace polysum { #define rep(i,a,n) for (int i=a;i<n;i++) #define per(i,a,n) for (int i=n-1;i>=a;i--) typedef long long ll; const ll mod=1e9+7; /// 取模值 ll powmod(ll a,ll b) {ll res=1;a%=mod; assert(b>=0); for(;b;b>>=1){if(b&1)res=res*a%mod;a=a*a%mod;}return res;} const int D=101000; /// 最高次限制 ll a[D],f[D],g[D],p[D],p1[D],p2[D],b[D],h[D][2],C[D]; ll calcn(int d,ll *a,ll n) { if (n<=d) return a[n]; p1[0]=p2[0]=1; rep(i,0,d+1) { ll t=(n-i+mod)%mod; p1[i+1]=p1[i]*t%mod; } rep(i,0,d+1) { ll t=(n-d+i+mod)%mod; p2[i+1]=p2[i]*t%mod; } ll ans=0; rep(i,0,d+1) { ll t=g[i]*g[d-i]%mod*p1[i]%mod*p2[d-i]%mod*a[i]%mod; if ((d-i)&1) ans=(ans-t+mod)%mod; else ans=(ans+t)%mod; } return ans; } void init(int M) { /// M:最高次 f[0]=f[1]=g[0]=g[1]=1; rep(i,2,M+5) f[i]=f[i-1]*i%mod; g[M+4]=powmod(f[M+4],mod-2); per(i,1,M+4) g[i]=g[i+1]*(i+1)%mod; } ll polysum(ll n,ll *arr,ll m) { // a[0].. a[m] \sum_{i=0}^{n-1} a[i] for(int i = 0; i <= m; i++) a[i] = arr[i]; a[m+1]=calcn(m,a,m+1); rep(i,1,m+2) a[i]=(a[i-1]+a[i])%mod; return calcn(m+1,a,n); } ll qpolysum(ll R,ll n,ll *a,ll m) { // a[0].. a[m] \sum_{i=0}^{n-1} a[i]*R^i if (R==1) return polysum(n,a,m); a[m+1]=calcn(m,a,m+1); ll r=powmod(R,mod-2),p3=0,p4=0,c,ans; h[0][0]=0;h[0][1]=1; rep(i,1,m+2) { h[i][0]=(h[i-1][0]+a[i-1])*r%mod; h[i][1]=h[i-1][1]*r%mod; } rep(i,0,m+2) { ll t=g[i]*g[m+1-i]%mod; if (i&1) p3=((p3-h[i][0]*t)%mod+mod)%mod,p4=((p4-h[i][1]*t)%mod+mod)%mod; else p3=(p3+h[i][0]*t)%mod,p4=(p4+h[i][1]*t)%mod; } c=powmod(p4,mod-2)*(mod-p3)%mod; rep(i,0,m+2) h[i][0]=(h[i][0]+h[i][1]*c)%mod; rep(i,0,m+2) C[i]=h[i][0]; ans=(calcn(m,C,n)*powmod(R,n)-c)%mod; if (ans<0) ans+=mod; return ans; } } int main() { polysum::init(N); while(~scanf("%d", &n)) { for (int i = 1; i <= n; ++i) scanf("%d", &a[i]); sort(a+1, a+1+n); LL pre = 1, ans = 0; for (int i = 1; i <= n; ++i) { b[0] = 0; for (int j = 1; j <= n-i+1; ++j) b[j] = (j*(polysum::powmod(j, n-i+1) - polysum::powmod(j-1, n-i+1))) % MOD; ans = (ans + pre*(polysum::polysum(a[i], b, n-i+1) - polysum::polysum(a[i-1], b, n-i+1))%MOD )%MOD; pre = (pre * a[i]) % MOD; } printf("%lld\n", (ans + MOD) % MOD); } return 0; }