I am interested in the sequence

$$a(n)=\sum_{k=0}^n {p(n-k)-1 \choose k}$$

where $p(n)=(r-1)n^2+(2r-1)n+r$ for some $r \in \mathbb{N}$ or more generally any polynomial equation.

When $r=1$ this reduces to the Fibonacci sequence which has exponential growth.

It can be seen that $a(n)$ has superexponential growth when $r \ge 2$ by considering the central term of the binomial sum $$a(n) \ge {p(n-(n/2))-1 \choose n/2}={p(n/2)-1 \choose n/2}\ge\left(\frac{p(n/2)-1}{n/2}\right)^{n/2}=\left(\sqrt{(r-1)\frac{n}{2}+(2r-1)+\frac{2n-2}{n}}\right)^n$$ 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 but I am not familiar enough with generating functions to extract any asymptotic information from it.