In [an answer](https://mathoverflow.net/a/322394/41291) to my question https://mathoverflow.net/q/320457/41291 (which was in turn caused by an older one, https://mathoverflow.net/q/146802/41291) I encountered the series
$$
F(t,q):=\sum_{n=1}^\infty(-1)^n\frac{q^nt}{\left(1-q^nt\right)^2}=-\sum_{n=1}^\infty\frac{n(qt)^n}{1+q^n}=\sum_{n=1}^\infty\left(\sum_{d|n}(-1)^{\frac nd}dt^d\right)q^n.
$$
It resembles closely several famous things like parametrization of the Tate curve, or $q$-expansions of theta functions. In particular, $F(t,q)+F(t^{-1},q)$ is "almost" the derivative of the logarithmic derivative of $\theta_2$, since
$$
\frac{d^2}{dz^2}\theta_2(z,q)=-\frac{t^2}{(1+t^2)^2}+8\sum(-1)^n\frac{nq^{2n}(t^{2n}+t^{-2n})}{1-q^{2n}};
$$ 
yet, I cannot go beyond that "almost" because of that silly plus sign in the denominator instead of minus.

What to make of this $F(t,q)$? Does it have some expression through known functions, or have some significance of its own?