The ratio in terms of sets The recurrence $a_{n+1}=a_n(n-1/2)$ is related  to $\Gamma(n+1/2)$ ( not difficult to prove) and it could be represented in a way like $\frac {(2n-1)!!} {2^n}$
Also I know that 
$(2n-1)!!$ is  the number of permutations of 2n whose cycle type consists of n parts equal to 2; these are the involutions without fixed points (A).
Also, for each $n \in N$, let $f(n)$ is  the number of subsets of set $[n]=\ {1,2,...,n}$. Then $f(n)=2^n$ (B)
I wonder about the understanding of the  meaning ( sense) of the ration: A/B?
What could be the meaning of $\frac {(2n-1)!!} {2^n}$ in terms of  sets?
 A: This is a good question, although its statement may need to be cleaned up a little bit. It occasionally happens that looking for patterns between seemingly different types of problems and seemingly different branches of mathematics produces real results. The issue with this particular problem is not that there aren't any interpretations that this quantity can be given, but that there are too many. The solution is not to throw this problem away, but to refine it to the point that it admits a clear answer. This is a natural process in mathematical research.
I suspect that your looking for a combinatorial interpretation to the formula

$\frac{\left(2n-1\right)!!}{2^n}$. Since $\mbox{gcd}\left(2^n,\left(2n-1\right)!!\right) = 1$ for all $n \geq 1$, this formula cannot be interpreted as enumerating the points in some specified finite set. Since $2^n < \left(2n-1\right)!!$ for all $n\geq 3$, this formula cannot be interpreted as a probability of some kind. 
The formula $\frac{\left(2n-1\right)!!}{n!2^n}$ can be interpreted as giving the probability that if two people each flip two separate fair coins $n$ times, then each person gets heads the same number of times. I worked this out by unraveling $\frac{A}{B}$ using binomial identities until I got something that looked like the probability of some easily described random event. 
Edit:
In the light of this question and its answers, it seems that there is a connection between this simple game I've described and the numbers you are interested in. It's strange how these things happen.
A: @Albert: The recurrence $M(n+1)/M(n)=n-1/2+o(1/n)$ is related to Kendall-Mann property
http://oeis.org/A181609
Could you look at the answer from Moron please
Recurrence representation(s): $a(n+1)=a(n)(n-1/2)+o(1/n)$ and $a(n+1)=a(n)(n-1/2+o(1/n))$
It seems to me that your game is a good interpretation, am I right?
