Here is a power series, which looks a bit like a Hypergeometric function series, but I don't think that it is. Has anyone any idea what it is? Here $n,p,r$ are integers with $n\ge 0$ and $p\ge r\ge 0$:

$$ f_{n,p,r}(x)\,=\,\sum_{s=0}^p \frac{x^s}{s!}\ \frac{p!\,(2n+p+s+2)!\,(n+r+s+2)!}{(p-s)!\,(2n+r+s+3)!\,(n+s+2)!} $$

Originally this occurred as a $q$-factorial series, but if the $q=1$ case given here could be recognised, it would be a big help.

Thanks for the answer below. A bit before I declare this answered (if I remember how to do that!) I would like to sneak in another question. Is there any sensible way to sum this in the case $x=-1$? The answer is likely zero if $r<p$ and reasonably simple if $r=p$, but how to prove that?