Finite summation

Question

I am struggling with following sum, can any body comment how I can attack this one? any bounds or approximations would help

\begin{equation} \sum_{i=0}^{n} i^2 a^{i^2} = a+ 4a^4 +9 a^9 +16 a^{16} +...+n^2 a^{n^2} \end{equation}