Let $e$ be a positive natural number, there is the following equality of formal power series $$\frac{1}{1-et}=\frac{\Pi_{i=1}^\infty(1+t^{2i-1})^{\alpha_{2i-1}}}{\Pi_{i=1}^\infty(1+t^{2i})^{\alpha_{2i}}}.$$
How can I use Möbius inversion to find the $\alpha$'s?