Let $T_n$ be the number of elements of $S_n$ with order $1$ or $2$. It is well known that:
$ T_n = \sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2k}(2k-1)!! = \frac{d^n}{dx^n}\left.\exp\left(x+\frac{x^2}{2}\right)\right|_{x=0}$
and, if we call $D_n = \frac{T_n}{n!},$ $ D_n = \frac{1}{n}\left(D_{n-1}+D_{n-2}\right). $
My question is: what is the asymptotic behaviour of $D_n$ as $n$ goes to infinity?
Is it possible to apply some saddle-point method in this case?