題目鏈接：點擊查看

題目大意：對于n>1，求的長度為n的數(shù)組個數(shù)

題目分析：因為長度為n的數(shù)組中，組成數(shù)字相同但排列組合不同的數(shù)組，不算同一種情況，所以我們在求出一種滿足條件的數(shù)組后，需要考慮所有排列組合的情況，然后枚舉的時候也有技巧，如果盲目枚舉3000位，那得枚舉到何年何月。。我們需要分析一下這個式子，可以得出一個簡單地結(jié)論，就是當n很大的時候，絕大部分都是1，為什么這么說呢，現(xiàn)在假設(shè)n>14，并且假設(shè)現(xiàn)在有14個大于1的數(shù)，然后其余的數(shù)都為1，因為2的14次方就已經(jīng)大于3000了，所以那14個大于1的數(shù)累乘起來必定是大于3000的，所以將這些元素累加起來的結(jié)果是絕對不可能滿足上述公式的。

現(xiàn)在已經(jīng)剪枝了絕大部分，因為大于1的元素最多有13個，所以當n小于13的時候枚舉n就行，當n大于13的時候，枚舉一下13個不同的元素，然后剩下的用1補齊，然后計算全排列即可，因為我數(shù)論賊菜，所以全排列公式是我現(xiàn)學現(xiàn)賣的。。舉個很簡單的例子，比如3000個數(shù)中，有5個2,3個4，其余的都是1，那么排列組合就是(3000!)/(2992!*5!*3!)，因為涉及到了除法，所以我們可以預(yù)處理一下每個全排列的逆元即可。

首先先說一下幾個小性質(zhì)，也是挺簡單地性質(zhì)，雖然我不會證明。。

每個枚舉的都是小于等于n的

所有之和小于等于2*n

在dfs中還需要剪枝的地方就是乘積了，因為在枚舉的時候，我選擇枚舉是不嚴格升序枚舉的，所以當累乘大于2*n時就可以直接返回了，并且枚舉每一位的時候，下限是上一次枚舉的數(shù)值，上限是2*n/mul (mul為累乘到當前的值)，因為要保證累乘的結(jié)果總是小于2*n，所以這樣枚舉也可以達到大幅降低時間復(fù)雜度的效果，下面掛上打表代碼，因為跑了不到半分鐘，然后如果是直接跑的話大概時間是3秒鐘，但是題目給的是1秒鐘，只能把3000個數(shù)值跑出來然后打表了。。

打表代碼：

#include<iostream> #include<string> #include<cstring> #include<cstdio> #include<algorithm> #include<climits> #include<cmath> #include<cctype> #include<stack> #include<queue> #include<list> #include<vector> #include<set> #include<map> using namespace std;typedef long long LL;const int inf=0x3f3f3f3f;const int mod=1e9+7;const int limit=13;const int N=3e3+100;LL q_pow(LL a,LL b) {LL ans=1;while(b){if(b&1)ans=ans*a%mod;a=a*a%mod;b>>=1;}return ans; }LL fac[N],ifac[N],cnt[N];LL ans;int n;void dfs(int pos=0,int pre=1,LL plus=0,LL mul=1)//pos：當前枚舉了多少位，pre：上一位是多少，plus：累加，mul：累乘 {if(pos==min(n,limit)){int t=n>limit?n-limit:0;//計算有在至多13個不同的數(shù)值中有多少個1if(plus+t!=mul)return;LL temp=fac[n];//按照公式實現(xiàn)temp=temp*ifac[cnt[1]+t]%mod;//實現(xiàn)。。for(int i=2;i<N;i++)//還是實現(xiàn)，不過這里很巧妙，因為ifac[0]=1，所有沒有出現(xiàn)的數(shù)x的cnt[x]都是0，所以讓temp*1答案還是temptemp=temp*ifac[cnt[i]]%mod;ans=(ans+temp)%mod;//更新最終答案return;}if(mul>2*n)//剪枝return;for(int i=pre;i<=2*n/mul;i++)//枚舉{cnt[i]++;dfs(pos+1,i,plus+i,mul*i);cnt[i]--;} }int main() {freopen("output.txt","w",stdout);fac[0]=ifac[0]=1;for(int i=1;i<N;i++){fac[i]=fac[i-1]*i%mod;//求全排列ifac[i]=q_pow(fac[i],mod-2);//求全排列的逆元}for(n=2;n<=3000;n++){ans=0;dfs();printf("%lld,\n",ans);}return 0; }

AC代碼：

#include<iostream> #include<string> #include<cstring> #include<cstdio> #include<algorithm> #include<climits> #include<cmath> #include<cctype> #include<stack> #include<queue> #include<list> #include<vector> #include<set> #include<map> #include<unordered_map> using namespace std;typedef long long LL;const int inf=0x3f3f3f3f;const int N=3e3+100;int 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main() { // freopen("output.txt","w",stdout);int w;for(cin>>w;w;w--){int n;scanf("%d",&n);printf("%d\n",ans[n]);}return 0; }

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