Let $$ A(x)=\sum_{j=0}^{n}\frac{x^{2j}}{j!},$$ $$B(x)=\sum_{j=0}^{n}j\frac{x^{2j-1}}{j!},$$
$$C(x)=\sum_{j=0}^{n}j^{2}\frac{x^{2j-2}}{j!},$$ and $$g_n(x)=\frac{1}{\pi }\frac{\sqrt{A(x)C(x)-(B(x))^{2}}}{A(x)}.$$
Show the asymptotics $$\int_{\mathbb{R}} g_n(x)dx=(\frac{2}{\pi }+o(1))\sqrt{n}.$$
Note this asymptotic is the expected number of real zeros for Flat Polynomials, and all these $A$, $B$, $C$ are the results of reproducing kernel.
I tried this by using simple algebra , after symplifying I obtain this expression for $g_n(x)$ : $$g_n(x)=\frac{\sqrt{\sum_{j\neq k}(k^2-jk)\frac{x^{2j+2k-2}}{j!k!}}}{\sum_{j}\frac{^{x^{2j}}}{j!}}.$$ But then I dont know how to continue. I was given a hint that, after some simple algebra, it is possible to use Stirling's formula to obtain this asymptotic. But How?