Chair Seating Probability

There is a long table with $n+2$ chairs arranged in a row and labeled from $0$ to $n+1$, with equal distances between any two adjacent chairs. Initially, one person sits on chair $0$ and another on chair $n+1$. Then, $m$ persons enter one by one and choose seats according to the following process:

1. They uniformly randomly choose an empty chair.
2. After sitting, they perform a sequence of moves. At each step, let the current position be $i$ and let $$d=\min_{x\in S}\;|i-x|,$$ where $S$ is the set of already occupied chairs. They check if moving to an adjacent (left or right) chair that is empty can increase $d$. If one or both adjacent moves yield a larger minimal distance, they move to the one with the larger value (in case of a tie, they choose the left move). This process repeats until no adjacent move can increase $d$.
   (It can be shown that this process terminates in a finite number of steps.)

For each chair numbered $1$ to $n$, determine the probability that it is eventually occupied by one of the $m$ persons. All results should be printed with 6 decimal places.

inputFormat

The input consists of a single line containing two integers $n$ and $m$ where $1\leq n \leq \text{small}$ and $m\geq 1$. The chairs are numbered from $0$ to $n+1$, and the initial occupied chairs are $0$ and $n+1$.

outputFormat

Output $n$ numbers separated by a space. The $i$-th number (for $1\leq i \leq n$) represents the probability that chair $i$ is eventually occupied, printed with 6 decimal places.

sample

1 1

1.000000