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I'm interested in two things:

  1. A computationally efficient (used here to mean the number of terms is bounded and not dependant on the size of the input) partial sum formulae of the expression below, and barring that:

  2. A closed form for:

$$\sum_{m \ge 1} \frac{z^m}{{(2m)!}(1-q^m)}$$ Please see user Jjacqueline's answer to a very similar question here: Scroll down to Jjacqueline's answer, I couldn't get the citation feature working for some reason

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