Writing generalized hypergeometric function ${_nF}_{n-1}(\cdot)$ as a function of $n$

How can we code the hypergeometric function $ $ f(n,k,z)={_nF}_{n-1}{\huge(}{1-k,\overbrace{2,\dots,2}^{n-1\ \text{times}} \atop \underbrace{1,\dots,1}_{n-1\ \text{times}}};z{\huge)} $ $ as a function that accepts any $ n=1,2,3,\dots$ ? This function reduces to a rational function in $ z$ but that is not what I’m after. I apologize for lack of an attempt here…did not have any idea where to start.