Is it possible to solve the following inequality explicitly? Or at least proving it with some method? $$3^n \frac{\pi^{n/2}}{(n/2)!}(1+e^{-\pi}) 2^{-\frac{3n}{2}} \le 2^{-n/2+8}$$ For $n \in \mathbb{N}$.
I am reading a research paper where they leave this up to the reader and I couldn't find a way to separate the variables, or even with Stirling's approximation. Also taking the derivative of the function results in a mess.
Graphing the functions makes the inequality seem plausible (the dots are $f(a)$, and the dashes are $f(b))$:
Hint: Your inequality is equivalent to: $$\dfrac{\pi^{n/2}}{(n/2)!}(1+e^{-\pi})\leq 256\left(\frac{2}{3}\right)^n$$ or $$f(n) = \dfrac{(2.25\cdot\pi)^{n/2}}{(n/2)!}\leq\dfrac{256}{1+e^{-\pi}}=245.395...$$
Now, you can simply observe that the LHS is eventually decreasing eventually because a factorial is increasing more rapidly than an exponent. This means there is a single, global maximum at some $n = k$, where it has to satisfy the inequalities: $$f(k-1)<f(k)$$ and $$f(k)>f(k+1).$$
If you solve these inequalities, then you can easily find that $k = 14$, which gives the maximum of $f(k) = 174.943.$