Let $a_n=[x^n]\mathrm{e}^{x+x^2/2}.$ How does one show that $ a_n \sim\frac{1}{2\sqrt{\pi}} n^{-(n+1)/2}\mathrm{e}^{-n/2+\sqrt n -1/4}?$
I'd also appreciate references illustrating relevant techniques.
Let $a_n=[x^n]\mathrm{e}^{x+x^2/2}.$ How does one show that $ a_n \sim\frac{1}{2\sqrt{\pi}} n^{-(n+1)/2}\mathrm{e}^{-n/2+\sqrt n -1/4}?$
I'd also appreciate references illustrating relevant techniques.
The function $ f(x) = e^{x+x^2/2} $ is the exponential generating function for the number of involutions on finite sets.
An analytic derivation of the asymptotic formula $ [x^n]f(x) = \frac{1}{2\sqrt{\pi}} n^{-(n+1)/2} e^{n/2+\sqrt{n}-1/4}\left(1+O\left(n^{-1/5}\right)\right) $ can be found on pages 558-560 in Flajolet and Sedgewick's Analytic Combinatorics (freely available here). Flajolet also cites volume 3 of Knuth's The Art of Computer Programming, which he says contains a derivation of the bound through the use of the explicit formula $ [x^n]f(x) = \sum_{k=0}^{\lfloor n/2 \rfloor} \frac{1}{(n-2k)!2^k k!}. $