The leading root of the partial theta function

I study the leading root x0(y) of the partial theta function , considered as a formal power series. I prove that all the coefficients of −x0(y) are strictly positive. Indeed, I prove the stronger results that all the coefficients of −1/x0(y) after the constant term 1 are strictly negative, and all the coefficients of 1/x02(y) after the constant term 1 are strictly negative except for the vanishing coefficient of y3.