C. Expected Final Score
time limit per test
1 second
memory limit per test
128 megabytes
input
standard input
output
standard output

You are given a set $$$S$$$ of integers, initially containing $$$n$$$ elements. Each element in $$$S$$$ is assigned a unique index from $$$1$$$ to $$$n$$$. You are also given an integer $$$p$$$ (where $$$0 \le p \le n$$$).

You will perform the following sequence of operations until the set $$$S$$$ becomes empty:

  • Select an index $$$i$$$ (where $$$1 \le i \le \text{current length of }S$$$).
  • Remove the element at index $$$i$$$ from $$$S$$$.
  • If $$$i \gt p$$$, decrease $$$p$$$ by $$$1$$$.
  • After the deletion, all elements in $$$S$$$ with indices greater than $$$i$$$ are shifted one position to the left.

The final value of $$$p$$$ after all deletions have been performed is the final score. What will be the expected final score if all operations are performed randomly?

You are given the value of $$$n$$$. For each possible initial value of $$$p$$$ (where $$$0 \le p \le n$$$), determine the expected final score after performing all deletions randomly.

Input

An integer $$$n$$$ ($$$1 \le n \le 10^3$$$), the initial number of elements in the set $$$S$$$.

Output

Print $$$n+1$$$ space-separated floating values, the expected final score after all elements have been deleted for each possible initial value of $$$p$$$ from $$$0$$$ to $$$n$$$.

Your answer will be considered correct if its absolute or relative error does not exceed $$$10^{-6}$$$. That is, if your answer is $$$a$$$, and the jury's answer is $$$b$$$, then the solution will be accepted if $$$\frac{|a-b|}{\max(1,|b|)} \le 10^{-6}$$$.

Example
Input
3
Output
-3 -1.3333333333 1.3333333333 3