This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then,
$$\alpha(\tau) = \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$
where $\eta(\tau)$ is the Dedekind eta function and coefficients are A007191.
For example, let $\tau =\sqrt{-1}$. Then $\alpha(\tau) = 512 = 2^9$ and the series $(1)$ "explains" why,
\begin{align} 512 &\approx e^{2\pi}-24\\[4pt] 512(1+\sqrt2)^3 &\approx e^{2\pi\sqrt2}-24 \end{align}
and so on for other $\tau$.
Q: Let $q = e^{-\pi}$. Can we find a relation, $$\pi = \frac{1}{q} - 20 +c_1 q + c_2 q^2 +c_3 q^3 +\cdots\tag2$$ where the $c_i$ are well-defined integers or rationals such that $(2)$ "explains" why $\pi \approx e^{\pi}-20$?
For example, we have the rather curious functions,
$$\beta_1(q) := \frac{1}{q} - 20 +\tfrac{1}{48}q - \tfrac{1}{300}q^3 -\tfrac{1}{972}q^5 +\tfrac{1}{2160}q^7+\tfrac{1}{\color{brown}{2841}}q^9-\tfrac{1}{\color{brown}{2369}}q^{11}-\cdots\tag3$$
$$\beta_2(q) := \frac{1}{q} - 20 +\tfrac{1}{48}q - \tfrac{1}{300}q^3 -\tfrac{1}{972}q^5 +\tfrac{1}{2160}q^7+\tfrac{1}{\color{brown}{2842}}q^9-\tfrac{1}{\color{brown}{2810}}q^{11}-\cdots\tag4$$
Let $q = e^{-\pi}$, then,
$$\beta_1(q) \approx \pi,\quad (\text{difference:}\; {-4}\times 10^{-22})\\ \beta_2(q) \approx \pi,\quad(\text{difference:}\; {-3}\times 10^{-22})$$
However, there seems to be an indefinite number of formulas, where the choice of a coefficient (say, $2841$ or $2842$) determines an ever-branching tree of formulas. But there might be a subset where the coefficients have a nice closed-form.
