So there's an elementary (but in my opinion quite interesting!) result which is that the laurent series expansion of

$$\frac{1}{1-e^x} = -\frac{1}{x} + \frac{1}{2} - \frac{1}{12}x - \frac{1}{720}x^3 \cdots$$

Now the reason that is interesting is because each of those coefficients are equal to $\frac{1}{n!}B_n$ where $B_n$ is $n$th bernoulli number and so more deeply one has the result

$$ \Gamma(\alpha + 1) \frac{d^\alpha}{dx^\alpha} \left[ \frac{1}{1-e^x} - 1 \right] = (-1)^{\alpha-1} \alpha \zeta(1-\alpha) $$

If you choose your fractional derivative carefully. ^^ (there might be an off by 1 error there, I need to double check)

So naturally I was inspired to go hunting for other elementary functions and to look at their coefficients/fractional derivatives in the hopes of finding something cool.

So I decided to look at $\frac{1}{1-e^{e^x-1}}$ the laurent series expansion of this is

$$\frac{1}{1-e^{e^x-1}} = -\frac{1}{x} + 1 - \frac{x}{6} - \frac{x^2}{24} - \frac{x^3}{90} - \frac{x^4}{720} + \frac{59}{60480}x^5 + \frac{17}{20160}x^6 + \frac{169}{453600} x^7 + \frac{47}{483840}x^8 - \frac{181}{119750400}x^9 \cdots$$

This was generated using this python code:

import sympy as sym
import math

x = sym.symbols('x')

w = 1/(1 - sym.exp(sym.exp(x)-1))
result = w.series(x, 0, 10).removeO()
print(result)

So I went over to OEIS and tried looking for those denominators and found nothing, I also looked at the denominators scaled by n! and similarly came up empty handed.

So this has me rather surprised since this is a pretty simple series to write down. My first question then that I want to ask is: is there some recurrence or functional equation that these coefficients/some scaled version of these coefficients obey?

If no recurrence then at least some kind of interpretation of their combinatorial significance would be cool.