Using exponential formula to find generating functions I have that $a_n$ is the number of permutations of $[n]$ where all cycles are odd length and $b_n$ is the number of permutations of $[n]$ with even number of cycles.
The exponential formula is described here.
I am trying to use the exponential formula to show that if $A(x)$ is the generating function for $a_n$ and $B(x)$ is the generating function for $b_n$, then $A(x)=(\frac{1+x}{1-x})^{\frac{1}{2}}$ and $B(x)=(1-x^2)^{\frac{1}{2}}$.  A hint is given to consider how the exponential formula changes when an even number of component structures are to be used.
I don't understand what the exponential formula means, especially what component and general labeled structures are.
 A: In the terminology of combinatorial species we have the two species
$$\mathcal{A} = \mathfrak{P}(\mathfrak{C}_{=1}(\mathcal{Z})
+\mathfrak{C}_{=3}(\mathcal{Z})
+\mathfrak{C}_{=5}(\mathcal{Z})
+\mathfrak{C}_{=7}(\mathcal{Z})
+\cdots)$$
which translates to the generating function
$$A(z) = \exp\left(\sum_{q\ge 0} \frac{z^{2q+1}}{2q+1}\right)
= \exp\log\frac{1}{1-z} 
\exp\left(-\sum_{q\ge 1} \frac{z^{2q}}{2q}\right)
\\= \frac{1}{1-z} \exp\left(-\frac{1}{2}\log\frac{1}{1-z^2}\right)
= \frac{\sqrt{1-z^2}}{1-z} = \sqrt{\frac{1+z}{1-z}}.$$
For the second part we need the species
$$\mathcal{B} = \mathfrak{P}(\mathcal{U}\mathfrak{C}_{=1}(\mathcal{Z})
+\mathcal{U}\mathfrak{C}_{=2}(\mathcal{Z})
+\mathcal{U}\mathfrak{C}_{=3}(\mathcal{Z})
+\mathcal{U}\mathfrak{C}_{=4}(\mathcal{Z})
+\cdots)$$
which gives the generating function
$$G(z, u) = \exp\left(u\log\frac{1}{1-z}\right)$$
so that
$$B(z) = \frac{1}{2} \exp\left(+\log\frac{1}{1-z}\right)
+ \frac{1}{2} \exp\left(-\log\frac{1}{1-z}\right)
\\ = \frac{1}{2} \frac{1}{1-z} + \frac{1}{2} (1-z).$$
This means that a permutation on zero elements consists of an even number of cycles and the permutation on one element consists of an odd number of cycles and when $n\gt 1$ exactly half of all permutations consist of an even number of cycles.
Remark. The first EGF yields 
$$1, 1, 3, 9, 45, 225, 1575, 11025, 99225, 893025, 9823275, 108056025,\ldots$$
which is OEIS A000246 which may be consulted for additional readings.
