I recently found the asymptotic expansion for $\frac{(2n)!!}{(2n-1)!!}$ to be $\sqrt{\pi n}$ by simplifying the double factorials and applying Stirling's formula. However, I was unable to find an asymptotic approximation for $\frac{(3n)!!!}{(3n-1)!!!}$ since triple factorials are much more difficult to work with.
I used the following multifactorial extensions to derive an asymptotic formula for $\frac{(3n)!!!}{(3n-1)!!!}$, however, I was unable to simplify this expression well enough to derive its asymptotic approximation.
Are there any well-known formulas for the $n^{th}$ factorial? Ideally, I want to find asymptotical approximations for $\frac{(kn)!_k}{(kn-1)!_k}$, where $!_k$ denotes the $k^{th}$ factorial. Any insight or alternative formulas for my calculations would be greatly appreciated!
Edit: I am assuming n is a positive integer for these calculations.
You can use the explicit formulae given in this answer together with the known asymptotic expansion for the ratio of two gamma functions to deduce $$ \frac{{(3n)!!!}}{{(3n-1)!!!}} = \frac{\Gamma\! \left( {\frac{2}{3}} \right)\Gamma (n + 1)}{{\Gamma\! \left( {n + \frac{2}{3}} \right)}} \sim \Gamma\! \left( {\tfrac{2}{3}} \right)n^{1/3} \left( {1 +\frac{1}{{9n}} - \frac{10}{{2187n^3 }} +\frac{11}{19683n^4}+ \ldots } \right) $$ as $n\to +\infty$. The method for higher analogues is similar.