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.
Best Answer
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.