It is not unique. Maple solves your differential equation in terms of HeunC functions:

$$w \! \left(t \right) = c_{1} \mathit{HeunC} \left(2 n -1, n -1, n -2, -n^{2}+\frac{1}{2}, \frac{n^{2}}{2}+\frac{1}{2}, t\right)+c_{2} t^{-n +1} \mathit{HeunC} \left(2 n -1, -n +1, n -2, -n^{2}+\frac{1}{2}, \frac{n^{2}}{2}+\frac{1}{2}, t\right)$$ The first solution $$ \mathit{HeunC} \left(2 n -1, n -1, n -2, -n^{2}+\frac{1}{2}, \frac{n^{2}}{2}+\frac{1}{2}, t\right)$$ is analytic in the open unit disk.

It is not unique. Maple solves your differential equation in terms of HeunC functions:

$$w \! \left(t \right) = c_{1} \mathit{HeunC} \left(2 n -1, n -1, n -2, -n^{2}+\frac{1}{2}, \frac{n^{2}}{2}+\frac{1}{2}, t\right)+c_{2} t^{-n +1} \mathit{HeunC} \left(2 n -1, -n +1, n -2, -n^{2}+\frac{1}{2}, \frac{n^{2}}{2}+\frac{1}{2}, t\right)$$ The first basic solution $$ \mathit{HeunC} \left(2 n -1, n -1, n -2, -n^{2}+\frac{1}{2}, \frac{n^{2}}{2}+\frac{1}{2}, t\right)$$ is analytic in the open unit disk.