Deltix Round, Spring 2021 E. Crypto Lights 组合数学 + 推公式

传送门

题意：

给你$n$个灯，每次可以打开一个灯，当连续的$k$个灯有至少两个灯开着的时候停止，问最终期望能打开多少灯。

思路：

由于不想打$latex$，所以手推了公式。

实现起来就很简单啦。
$ans$初始为$1$是因为$i=0$的时候概率显然为$1$。

// Problem: E. Crypto Lights
// Contest: Codeforces - Deltix Round, Spring 2021 (open for everyone, rated, Div. 1 + Div. 2)
// URL: https://codeforces.com/problemset/problem/1523/E
// Memory Limit: 256 MB
// Time Limit: 3000 ms
// 
// Powered by CP Editor (https://cpeditor.org)

//#pragma GCC optimize("Ofast,no-stack-protector,unroll-loops,fast-math")
//#pragma GCC target("sse,sse2,sse3,ssse3,sse4.1,sse4.2,avx,avx2,popcnt,tune=native")
//#pragma GCC optimize(2)
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#define X first
#define Y second
#define L (u<<1)
#define R (u<<1|1)
#define pb push_back
#define mk make_pair
#define Mid (tr[u].l+tr[u].r>>1)
#define Len(u) (tr[u].r-tr[u].l+1)
#define random(a,b) ((a)+rand()%((b)-(a)+1))
#define db puts("---")
using namespace std;

//void rd_cre() { freopen("d://dp//data.txt","w",stdout); srand(time(NULL)); }
//void rd_ac() { freopen("d://dp//data.txt","r",stdin); freopen("d://dp//AC.txt","w",stdout); }
//void rd_wa() { freopen("d://dp//data.txt","r",stdin); freopen("d://dp//WA.txt","w",stdout); }

typedef long long LL;
typedef unsigned long long ULL;
typedef pair<int,int> PII;

const int N=1000010,mod=1e9+7,INF=0x3f3f3f3f;
const double eps=1e-6;

int n,k;
LL fun[N],inv[N];

LL qmi(LL a,LL b) {
	LL ans=1;
	while(b) {
		if(b&1) ans=ans*a%mod;
		a=a*a%mod;
		b>>=1;
	}
	return ans%mod;
}

void init() {
	fun[0]=1;
	for(int i=1;i<N;i++) fun[i]=fun[i-1]*i%mod;
	inv[N-1]=qmi(fun[N-1],mod-2);
	for(int i=N-2;i>=0;i--) inv[i]=inv[i+1]*(i+1)%mod;
}

LL C1(int n,int m) {
	return fun[n]*inv[n-m]%mod*inv[m]%mod;
}

LL C2(int n,int m) {
	return fun[n-m]*fun[m]%mod*inv[n]%mod;
}

int main()
{
//	ios::sync_with_stdio(false);
//	cin.tie(0);	

	init();
	int _; scanf("%d",&_);
	while(_--) {
		scanf("%d%d",&n,&k);
		LL ans=1;
		for(int i=1;i<=n-1;i++) if(n-1ll*(i-1)*(k-1)>=i) ans+=C1(n-1ll*(i-1)*(k-1),i)*C2(n,i)%mod,ans%=mod;
		printf("%lld\n",ans%mod);
	}




	return 0;
}
/*

*/