Entire function with $f(n) = \mathrm{sign}(n)\!\cdot\!n^2$ for nonzero integer $n$.

Question

To rephrase, I'm searching for an entire function with $f(n) = n^2$ and $f(-n) = -n^2$ for all positive integers $n$. I found this question on a Ph.D. qualifying exam, here (Q8)

https://math.unm.edu/sites/default/files/files/qual-exams/complex/unm_exam_201508_cmpx_qual.pdf

Show that there exists an entire function $~f(z)~$ such that $~f(n) = n^2~$ at every positive integer $~n~$ and $~f(n) = −n^2~$ at every negative integer $~n.~$ How many such entire functions are there?

and I'm a bit stumped by it. Can anyone give a hint?

Greg