Number of elements of order $2$ in $S_n$ 
How many elements of order $2$ are there in $S_n$?

Using combinatorics I arrived at this:
For $n$ even ($n=2k$) there are ${n\choose2}+{n\choose 2}{n-2\choose 2}\dfrac{1}{2!}+{n\choose 2} {n-2\choose 2}{n-4\choose 2}\dfrac{1}{3!}+\cdots+{n\choose 2}{n-2\choose 2}{n-4\choose 2}\cdots{2\choose 2}\dfrac{1}{k!}$.
For $n$ odd ($n=2k+1$) there are ${n\choose 2}+{n\choose 2}{n-2\choose 2}\dfrac{1}{2!}+{n\choose 2}{n-2\choose 2}{n-4\choose 2}\dfrac{1}{3!}+\cdots+{n\choose 2}{n-2\choose 2}{n-4\choose 2}\cdots{3\choose 2}\dfrac{1}{k!}$
But how do I find the sums?
Seems like I have to use induction. But not quite upto there.
Thanks for the help!!

it is probably not any help, but i think your sum may be written $f(\sqrt{2})$ where
$$
f(x) = \sqrt{2}^{-n} \sum_{k=1}^{\lfloor \frac{n}2 \rfloor} \frac1{k!}\frac{d^{2k}}{dx^{2k}}x^n
$$
