Closed form for $\sum_{k=0}^{n}{{n \choose k}a^{k^2}}$

Question

I think the following sum:

$$\sum_{k=0}^{n}{{n \choose k}a^{k^2}}$$

could be computed recursively using the fact that $(k+1)^2 = k^2+2k+1$, but do somebody know some reference work about that problem, or maybe even its generalization for any positive integer $m$:

$$\sum_{k=0}^{n}{{n \choose k}a^{k^m}}$$