How does $\tbinom{4n}{2n}$ relate to $\tbinom{2n}{n}$?

Question

I got this question in my mind when I was working on a solution to factorial recurrence and came up with this recurrence relation: $$(2n)!=\binom{2n}{n}(n!)^2$$ which made me wonder: is there also a recurrence relation for $\tbinom{4n}{2n}$ in terms of $\tbinom{2n}{n}$? Please use no factorials greater than $(2n)!$, preferably not greater than $n!$.

Answer 1

There can't be any multiplicative formula of the sort you describe for $4n\choose 2n$ in terms of $2n\choose n$, because there are prime factors of the former that aren't factors of the latter. By Bertrand's Postulate there's a prime number between $k=2n$ and $2k$($=4n$); that prime will be a factor of $4n\choose 2n$, but can't be a factor of $2n\choose n$ or any factorial of the form $(2n)!$ or less.

Answer 2

I think this is what you are looking for (but this is just back substitution):

$$\binom{4n}{2n} = \frac{(4n)!}{((2n)!)^2} = \frac{(4n)!}{\binom{2n}{n}^2 (n!)^4}.$$

Please update me on whether this is what you're looking for...

Answer 3

Here is an estimate that gives a good approximation of $\binom{4n}{2n}$ in terms of $\binom{2n}{n}$.

Using the identity $$ (2n-1)!!=\frac{(2n)!}{2^nn!}\tag{1} $$ it is straightforward to show that $$ \frac{\binom{4n}{2n}}{\binom{2n}{n}}=\frac{(4n-1)!!}{(2n-1)!!^2}\tag{2} $$ Notice that $$ \begin{align} \frac{(2n-1)!!}{2^nn!} &=\frac{2n-1}{2n}\frac{2n-3}{2n-2}\frac{2n-5}{2n-4}\cdots\frac12\\ &=\frac{n-\frac12}{n}\frac{n-\frac32}{n-1}\frac{n-\frac52}{n-2}\cdots\frac{\frac12}{\;1}\\ &=\frac1{\sqrt\pi}\frac{\Gamma(n+\frac12)}{\Gamma(n+1)}\tag{3} \end{align} $$ By Gautschi's Inequality, we have $$ \frac1{\sqrt{n+1}}\le\frac{\Gamma(n+\frac12)}{\Gamma(n+1)}\le\frac1{\sqrt{n}}\tag{4} $$ Thus, $(3)$ and $(4)$ yield $$ \frac1{\sqrt{\pi(n+1)}}\le\frac{(2n-1)!!}{2^nn!}\le\frac1{\sqrt{\pi n}}\tag{5} $$ and $$ \frac1{\sqrt{\pi(2n+1)}}\le\frac{(4n-1)!!}{2^{2n}(2n)!}\le\frac1{\sqrt{\pi 2n}}\tag{6} $$ Dividing $(6)$ by $(5)$ gives $$ \sqrt{\frac{n}{2n+1}}\le\frac{(4n-1)!!}{(2n-1)!!^2}4^{-n}\le\sqrt{\frac{n+1}{2n}}\tag{7} $$ Combine $(2)$ and $(7)$ to get $$ 4^n\sqrt{\frac{n}{2n+1}}\le\frac{\binom{4n}{2n}}{\binom{2n}{n}}\le4^n\sqrt{\frac{n+1}{2n}}\tag{8} $$ or asymptotically $$ \binom{4n}{2n}\sim\frac{4^n}{\sqrt2}\binom{2n}{n}\tag{9} $$