Integrating

$$y^{-2}(1-y^{n})^{-1/2}=\sum_{k=0}^{\infty}\frac{(\frac{1}{2})_{k}}{k!}y^{nk-2},$$

and utilizing the initial conditions to fix the leading coefficients,

$$g(y)=\sum_{k=0}^{\infty}\frac{(\frac{1}{2})_{k}}{k!}\frac{y^{nk}}{nk-1}.$$

The formula for $g(y)$ follows from the observation $\frac{-1}{nk-1}=\frac{(-\frac{1}{n})_{k}}{(\frac{n-1}{n})_{k}}.$ The evaluation of $g(1)$ appeals to Gauss’ hypergeometric theorem [1, Section 1.3]:

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