Finite sum of factorials, containing e, pi and bessel function

Question

In I.S. Gradshteyn and I.M. Rhyzhik's table of integrals, series and products, an interesting sum is given on page 2, formula 0.126:
\begin{equation} \sum_{k=0}^{n}\frac{(n+k)!}{k!(n-k)!}=\sqrt{\frac{e}{\pi}}K_{n+\frac{1}{2}}\left(\frac{1}{2}\right) \end{equation} Where $K_v(z)$ are the Bessel functions of imaginary argument, which are equal to the integral \begin{equation} K_v(z)=\int_{0}^{\infty}e^{-z\cosh t}\cosh vt\,dt \end{equation} How can one prove the sum? You can reference the article containing the proof.

Answer 1

In fact, the first results comes from the fact that $$\sum_{k=0}^{n}\frac{(n+k)!}{k!(n-k)!}x^k=\frac{e^{\frac{1}{2 x}}}{\sqrt{\pi x} } K_{n+\frac{1}{2}}\left(\frac{1}{2 x}\right)$$

For $x=1$, this generates the sequence $$\{1,3,19,193,2721,49171,1084483,28245729,848456353,28875761731,\cdots\}$$ which is $A001517$ in $OEIS$.

In the formula section, you could see an asymptotics proposed by Vaclav Kotesovec in $2014$

$$a_n \sim 2^{2 n+\frac{1}{2}}\, e^{\frac{1}{2}-n}\, n^n$$ For $n=10$, this gives

$$a_{9}\sim\frac{101559956668416 \sqrt{2}}{e^{17/2}}=29223646682$$

The relative error is less than $1.00$% if $n >10$, less than $0.10$% if $n> 105$ and less than $0.01$% if $n>1042$.