[洛谷P5081]Tweetuzki 爱取球

题目大意：有$n$个球，每一次取一个球然后放回，问期望多少次取遍所有球

题解：令$f_i$表示已经取了$i$种球，还要取的次数的期望。$f_i=\dfrac in(f_i+1)+\dfrac{n-i}n(f_{i+1}+1),f_n=0$

解个方程可得$f_i=\dfrac n{n-i}+f_{i+1}$，所有答案为$n\sum\limits_{i=1}^n\dfrac 1i$

卡点：无

C++ Code：

#include <cstdio>
#define maxn 10000010
const int mod = 20040313;
#define mul(x, y) static_cast<long long> (x) * (y) % mod

inline void reduce(int &x) { x += x >> 31 & mod; }

int n, ans;
int inv[maxn];
int main() {
	scanf("%d", &n);
	inv[0] = inv[1] = ans = 1;
	for (int i = 2; i <= n; ++i) {
		inv[i] = mul(mod - mod / i, inv[mod % i]);
		reduce(ans += inv[i] - mod);
	}
	printf("%lld\n", mul(ans, n));
	return 0;
}