I am interested in the sequence

$$a(n)=\sum_{k=0}^n {p(n-k) \choose k}$$

where $p(n)$ is a polynomial equation.

When $p(n)=n$ this reduces to the Fibonacci sequence, but what about when $p(n)$ is quadratic?

For example when $p(n)=n^2$, it can be seen that $a(n)$ has superexponential growth by considering only one of the terms of the sum $$a(n) \ge {p(n-(n/2)) \choose n/2}={p(n/2) \choose n/2}\ge\left(\frac{p(n/2)}{n/2}\right)^{n/2}=\left(\sqrt{\frac{n}{2}}\right)^n$$ But I would like to know more information than just this lower bound - an asymptotic formula would be great. Any ideas?

I found a related sequence here (which is equivalent to the case when $p(n)=n^2$) along with its generating function if that is any help to anyone.