Asymptotic expansions of zeros of a partial theta function

The bivariate series $\theta (q,x):={\sum }_{j=0}^{\mathrm{\infty }}{q}^{j(j+1)\mathrm{/}2}{x}^j$ defines a {\em partial theta function}. For fixed $q$ ($\mathrm{\mid }q\mathrm{\mid }<1$), $\theta (q,\mathrm{.})$ is an entire function. We prove a property of stabilization of the coefficients of the Laurent series in $q$ of the zeros of $\theta$. The coefficients $r_k$ of the stabilized series are positive integers. They are the elements of a known increasing sequence satisfying the recurrence relation $r_k=\sum {\nu =1}^{\infty}(-1)^{\nu -1}(2\nu +1)r{k-\nu (\nu +1)/2}$.