链接：
https://www.nowcoder.com/acm/contest/139/F
题意：
分析：
转载自：http://tokitsukaze.live/2018/07/19/2018niuke1.F/
代码：
#include <cstdio>
#include <cassert>
#include <algorithm>
using namespace std; /// 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+; /// 取模值
ll powmod(ll a,ll b) {ll res=;a%=mod; assert(b>=); for(;b;b>>=){if(b&)res=res*a%mod;a=a*a%mod;}return res;} const int D=; /// 最高次限制
ll a[D],f[D],g[D],p[D],p1[D],p2[D],b[D],h[D][],C[D];
ll calcn(int d,ll *a,ll n) {
if (n<=d) return a[n];
p1[]=p2[]=;
rep(i,,d+) {
ll t=(n-i+mod)%mod;
p1[i+]=p1[i]*t%mod;
}
rep(i,,d+) {
ll t=(n-d+i+mod)%mod;
p2[i+]=p2[i]*t%mod;
}
ll ans=;
rep(i,,d+) {
ll t=g[i]*g[d-i]%mod*p1[i]%mod*p2[d-i]%mod*a[i]%mod;
if ((d-i)&) ans=(ans-t+mod)%mod;
else ans=(ans+t)%mod;
}
return ans;
}
void init(int M) { /// M：最高次
f[]=f[]=g[]=g[]=;
rep(i,,M+) f[i]=f[i-]*i%mod;
g[M+]=powmod(f[M+],mod-);
per(i,,M+) g[i]=g[i+]*(i+)%mod;
}
ll polysum(ll n,ll *arr,ll m) { // a[0].. a[m] \sum_{i=0}^{n-1} a[i]
for(int i = ; i <= m; i++) a[i] = arr[i];
a[m+]=calcn(m,a,m+);
rep(i,,m+) a[i]=(a[i-]+a[i])%mod;
return calcn(m+,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==) return polysum(n,a,m);
a[m+]=calcn(m,a,m+);
ll r=powmod(R,mod-),p3=,p4=,c,ans;
h[][]=;h[][]=;
rep(i,,m+) {
h[i][]=(h[i-][]+a[i-])*r%mod;
h[i][]=h[i-][]*r%mod;
}
rep(i,,m+) {
ll t=g[i]*g[m+-i]%mod;
if (i&) p3=((p3-h[i][]*t)%mod+mod)%mod,p4=((p4-h[i][]*t)%mod+mod)%mod;
else p3=(p3+h[i][]*t)%mod,p4=(p4+h[i][]*t)%mod;
}
c=powmod(p4,mod-)*(mod-p3)%mod;
rep(i,,m+) h[i][]=(h[i][]+h[i][]*c)%mod;
rep(i,,m+) C[i]=h[i][];
ans=(calcn(m,C,n)*powmod(R,n)-c)%mod;
if (ans<) ans+=mod;
return ans;
}
} typedef long long int LLI;
const LLI MOD = polysum::mod;
const int UP = 1e3 + ;
LLI a[UP], b[UP]; int main() {
polysum::init(UP);
int n;
while(~scanf("%d", &n)) {
for(int i = ; i <= n; i++) scanf("%lld", &a[i]);
sort(a+, a+n+);
LLI ans = , prod = ;
for(int i = ; i <= n; i++) {
if(a[i] == a[i-]) {
prod = prod * a[i] % MOD;
continue;
}
for(int x = ; x <= n-i+; x++) {
b[x] = (polysum::powmod(x, n-i+) - polysum::powmod(x-, n-i+) + MOD) % MOD * x % MOD;
}
LLI temp = (polysum::polysum(a[i]+, b, n-i+) - polysum::polysum(a[i-]+, b, n-i+) + MOD) % MOD;
ans = (ans + prod * temp % MOD) % MOD;
prod = prod * a[i] % MOD;
}
printf("%lld\n", ans);
}
return ;
}
拉格朗日插值法模板（杜教版）：
/// 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+; /// 取模值
ll powmod(ll a,ll b) {ll res=;a%=mod; assert(b>=); for(;b;b>>=){if(b&)res=res*a%mod;a=a*a%mod;}return res;} const int D=; /// 最高次限制
ll a[D],f[D],g[D],p[D],p1[D],p2[D],b[D],h[D][],C[D];
ll calcn(int d,ll *a,ll n) {
if (n<=d) return a[n];
p1[]=p2[]=;
rep(i,,d+) {
ll t=(n-i+mod)%mod;
p1[i+]=p1[i]*t%mod;
}
rep(i,,d+) {
ll t=(n-d+i+mod)%mod;
p2[i+]=p2[i]*t%mod;
}
ll ans=;
rep(i,,d+) {
ll t=g[i]*g[d-i]%mod*p1[i]%mod*p2[d-i]%mod*a[i]%mod;
if ((d-i)&) ans=(ans-t+mod)%mod;
else ans=(ans+t)%mod;
}
return ans;
}
void init(int M) { /// M：最高次
f[]=f[]=g[]=g[]=;
rep(i,,M+) f[i]=f[i-]*i%mod;
g[M+]=powmod(f[M+],mod-);
per(i,,M+) g[i]=g[i+]*(i+)%mod;
}
ll polysum(ll n,ll *arr,ll m) { // a[0].. a[m] \sum_{i=0}^{n-1} a[i]
for(int i = ; i <= m; i++) a[i] = arr[i];
a[m+]=calcn(m,a,m+);
rep(i,,m+) a[i]=(a[i-]+a[i])%mod;
return calcn(m+,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==) return polysum(n,a,m);
a[m+]=calcn(m,a,m+);
ll r=powmod(R,mod-),p3=,p4=,c,ans;
h[][]=;h[][]=;
rep(i,,m+) {
h[i][]=(h[i-][]+a[i-])*r%mod;
h[i][]=h[i-][]*r%mod;
}
rep(i,,m+) {
ll t=g[i]*g[m+-i]%mod;
if (i&) p3=((p3-h[i][]*t)%mod+mod)%mod,p4=((p4-h[i][]*t)%mod+mod)%mod;
else p3=(p3+h[i][]*t)%mod,p4=(p4+h[i][]*t)%mod;
}
c=powmod(p4,mod-)*(mod-p3)%mod;
rep(i,,m+) h[i][]=(h[i][]+h[i][]*c)%mod;
rep(i,,m+) C[i]=h[i][];
ans=(calcn(m,C,n)*powmod(R,n)-c)%mod;
if (ans<) ans+=mod;
return ans;
}
}