My question is exactly as in the title:
What is the generating function or closed form solution to the sum of inverse products of parts in all compositions of $n$?
This question was inspired by just browsing around OEIS sequences. Here is the OEIS sequence for the numerators and denominators of the sum of inverse products of parts in all compositions of $n$:
The progress that I have made is finding the solution to the sum of products of parts in all compositions of $n$, the solution was inspired by this thread:
Number of parts equal to $~1~$ in all compositions of $~n~$
But I am still unclear how to solve this problem.
Please help.