Recall the classical $\theta(q):=\prod_{k=1}^{\infty}(1-q^k)$ and define the sequences $a_n$ and $b_n$ by

$$\frac{\theta^3(q)}{\theta(q^3)}=\sum_{n=0}^{\infty}a_nq^n \qquad \text{and} \qquad F(q):=\sum_{i,j\in\Bbb{Z}}q^{i^2+ij+j^2}=\sum_{n=0}^{\infty}b_nq^n.$$

**Edit.** In accord with Noam's commentary, we may replace $\theta$ by $\eta$.

Question.Is the following true? If so, any proof?

$$b_n=\begin{cases} \,\,\,\,\,\,\, a_n\qquad \text{if $a_n\geq0$} \\ -2a_n \qquad \text{if $a_n<0$}. \end{cases}$$