Considering the Jacobi theta: $\theta_3(z) = \sum_{n\in\mathbb{Z}} q^{n^2}$, we can invert $\theta_3-1$ in a small enough neighbourhood of 0.

Routine computation with Lagrange-Burmann inversion gives that the inverse have expansion starting by:

$\frac{q}{2}-\frac{q^4}{16}+\frac{q^7}{32}-\frac{q^9}{512}-\frac{11 q^{10}}{512}+\frac{13 q^{12}}{4096}+ O(q^{13})$

Is there any known closed form (or recursive formula) giving the terms of this series?

(it is very closely related to OEIS A259938 suite: https://oeis.org/A259938, but which doesn't seem very studied as well)