Approximate formula involving binomial coeffcients

Question

I am trying to find an approximate expression for the following formula $$f(m) = 2^m \binom{\frac{n+m}{2}}{m}\binom{n+m}{m}^{-1},$$ where $n$ is fixed and large and $n+m$ are integers with the same parity (both even or odd) and $m = 0,\ldots, n$. When I plot it for $n=400$ it looks like a Gaussian function centered at $m=0$ with $\sigma = \sqrt{n}$. Any idea how to analytically find the approximate formula for large $n$?

Answer 1

Allright, let's see. We'll use ${i\choose j}=\frac{i!}{j!(i-j)!}$. This makes:

\begin{align} f(m)&=2^m \binom{\frac{n+m}{2}}{m}\binom{n+m}{m}^{-1}\\ &=2^m\frac{((n+m)/2)!}{m!((n-m)/2)!}\cdot\frac{m!n!}{(m+n)!}\\ &=2^m\frac{((n+m)/2)!n!}{(m+n)!((n-m)/2)!} \end{align}

Since $n!\sim n^ne^{-n}\sqrt{2\pi n}$ by Stirling's Approximation, we can substitute that to approximate $f(m)$;

\begin{align} f(m)&=2^m\frac{((n+m)/2)!n!}{(m+n)!((n-m)/2)!}\\ &\approx 2^m\frac{((\tfrac12(n+m))^{(n+m)/2}e^{-(n+m)/2}\sqrt{\pi (n+m)}n^ne^{-n}\sqrt{2\pi n}}{((n+m)^{n+m}e^{-n-m}\sqrt{2\pi (n+m)}((\tfrac12(n-m))^{(n-m)/2}e^{-(n-m)/2}\sqrt{\pi (n-m)}}\\ &=\frac{n^n\sqrt{n}}{(n+m)^{(n+m)/2}(n-m)^{(n-m)/2}\sqrt{n-m}}\\ &=\sqrt{\frac{n^{2n+1}}{(n+m)^{n+m}(n-m)^{n-m+1}}} \end{align}

We can see a lot gets cancelled. I don't think we can simplify this even further.

If you wish though, you can create an upper and lower bound using this exact method; since we have $\sqrt{2\pi}\cdot\sqrt{n}n^ne^{-n}\leq n!\leq e\cdot \sqrt{n}n^ne^{-n}$, but I'll leave that up to you should you need that.

Answer 2

Another way to reason, pretty advanced though, is to use Gamma function together with a Taylor series at infinity:

$$\binom{\frac{n+m}{2}}{m} \sim \left(\frac{1}{n}\right)^{-m} \left(\frac{2^{-m}}{m \Gamma (m)}+\frac{2^{-m}}{\Gamma (m) n}+O\left(\left(\frac{1}{n}\right)^2\right)\right)$$

$$\binom{n+m}{m}^{-1} \sim \left(\frac{1}{n}\right)^m \left(m \Gamma (m)+\frac{-\frac{\Gamma (m) m^3}{2}-\frac{\Gamma (m) m^2}{2}}{n}+O\left(\left(\frac{1}{n}\right)^{3/2}\right)\right)$$