题目链接：uva 10288 - Coupons

题目大意：给定n，为有n中兑换卷，现在每开一次箱子，就能等概率的获得其中的一种兑换卷。问说平均情况下需要开多少个箱子才能集齐n种兑换卷。

解题思路：dp[i]表示还有i种没获得，dp[i]=n?in?dp[i]+in?dp[i?1]+1
===》dp[i]=dp[i?1]+ni

#include <cstdio>
#include <cstring>
#include <algorithm>

using namespace std;

typedef long long type;

struct Fraction {
    type member; // 分子；
    type denominator; // 分母；

    Fraction (type member = 0, type denominator = 1);
    void operator = (type x) { this->set(x, 1); }
    Fraction operator * (const Fraction& u);
    Fraction operator / (const Fraction& u);
    Fraction operator + (const Fraction& u);
    Fraction operator - (const Fraction& u);

    void set(type member, type denominator);
};

inline type gcd (type a, type b) {
    return b == 0 ? a : gcd(b, a % b);
}

/*Code*/
/////////////////////////////////////////////////////
const int maxn = 33;
typedef long long ll;

Fraction dp[maxn+5][maxn+5];

void init () {
    for (int i = 1; i <= maxn; i++) {
        dp[i][0] = 0;
        for (int j = 1; j <= i; j++)
            //dp[i][j] = (dp[i][j-1] * Fraction(j, i) + Fraction(1, 1)) / Fraction(j, i);
            dp[i][j] = dp[i][j-1] + Fraction(i, j);
    }
}

inline int countbit (ll u) {
    int ret = 0;
    while (u) {
        ret++;
        u /= 10;
    }
    return ret;
}


void put_ans (Fraction u) {

    type d = u.member / u.denominator;
    type member = u.member % u.denominator;

    int cn = countbit(d);
    int cd = countbit(u.denominator);

    int n = cn + (cn == 0 ? 0 : 1);
    if (member) {
        for (int i = 0; i < n; i++) printf(" ");
        printf("%lld\n", member);
    }

    printf("%lld", d);
    if (member) {
        if (d)
            printf(" ");
        for (int i = 0; i < cd; i++) printf("-");
        printf("\n");

        for (int i = 0; i < n; i++) printf(" ");
        printf("%lld", u.denominator);
    }
    printf("\n");
}

int main () {
    int n;
    init();
    while (scanf("%d", &n) == 1) {
        put_ans(dp[n][n]);
    }
    return 0;
}

/////////////////////////////////////////////////////

Fraction::Fraction (type member, type denominator) {
    this->set(member, denominator);
}

Fraction Fraction::operator * (const Fraction& u) {
    return Fraction(member * u.member, denominator * u.denominator);
}

Fraction Fraction::operator / (const Fraction& u) {
    return Fraction(member * u.denominator, denominator * u.member);
}

Fraction Fraction::operator + (const Fraction& u) {
    return Fraction(member * u.denominator + denominator * u.member, denominator * u.denominator);
}

Fraction Fraction::operator - (const Fraction& u) {
    return Fraction(member * u.denominator - denominator * u.member, denominator * u.denominator);
}

void Fraction::set (type member, type denominator) {
    type tmp_d = gcd(member, denominator);
    this->member = member / tmp_d;
    this->denominator = denominator / tmp_d;
}