The 2020 ICPC Asia Macau Regional Contest L - Random Permutation（概率期望/思维）

An integer sequence with length 𝑛n, denoted by 𝑎1,𝑎2,⋯,𝑎𝑛a1,a2,⋯,an, is generated randomly, and the probability of being 1,2,⋯,𝑛1,2,⋯,n are all 1𝑛1n for each 𝑎𝑖ai (𝑖=1,2,⋯,𝑛)(i=1,2,⋯,n).

Your task is to calculate the expected number of permutations 𝑝1,𝑝2,⋯,𝑝𝑛p1,p2,⋯,pn from 11 to 𝑛nsuch that 𝑝𝑖≤𝑎𝑖pi≤ai holds for each 𝑖=1,2,⋯,𝑛i=1,2,⋯,n.

Input

The only line contains an integer 𝑛n (1≤𝑛≤50)(1≤n≤50).

Output

Output the expected number of permutations satisfying the condition. Your answer is acceptable if its absolute or relative error does not exceed 10−910−9.

Formally speaking, suppose that your output is 𝑥x and the jury's answer is 𝑦y. Your output is accepted if and only if |𝑥−𝑦|max(1,|𝑦|)≤10−9|x−y|max(1,|y|)≤10−9.

Examples

input

Copy

2

output

Copy

1.000000000000

input

Copy

3

output

Copy

1.333333333333

input

Copy

50

output

Copy

104147662762941310907813025277584020848013430.758061352192

首先易知可能的a序列总共有\(n^n\)个。p序列一共有\(n!\)种可能（因为是1～n排列），对于每种排列，满足条件的a序列有\(n!\)个，因为对于\(p_1,p_2...p_n\)，a序列只能这样选：\([p1, n], [p2, n]...[pn, n]\)，且p1到pn是一个1到n的排列。注意因为是求期望，所以不用考虑重复的情况。

import math
n = eval(input())
print((math.factorial(n)) ** 2 / n ** n)