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.

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.

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

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

Now the reason that is interesting is because each of those coefficients are equal to $\frac{1}{n!}B_n$ where $B_n$ is nth 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

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

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.

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.

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

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

Now the reason that is interesting is because each of those coefficients are equal to $\frac{1}{n!}B_n$ where $B_n$ is nth 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

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

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?

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

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

Now the reason that is interesting is because each of those coefficients are equal to $\frac{1}{n!}B_n$ where $B_n$ is nth 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

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

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.