I have an infinite series that I need to write a function for (in Python) to calculate. To do so, I need to find the recursive definition of the formula, which I am having trouble with.

$$\frac{(-1)^nz^{2n+1}}{n!(2n+1)}$$

So far I have figured out the numerator to be

$$ t\_{n+1} = t\_n \times -z^2 $$

and the $n!$ turns into $(n+1)$, but I have no idea what the rest of the denominator turns into.

Any help would be much appreciated!