I'm having trouble in understanding how to use the sum in the relation below: $$ {}_sP_{jm}(\cos\theta) = \frac{(j+m)!}{\!\sqrt{(j+s)!(j-s)!}} \biggl(\!\sin{\frac{\theta}{2}}\biggr)^{\!2j} \, \sum_{n}^{j-s} \binom{j-s}{n} \binom{j+s}{n-m+s} (-1)^{j-n-s} \biggl(\!\cot{\frac{\theta}{2}}\biggr)^{\!2n-m+s}\,, $$ because when I try the ${}_\frac{1}{2}P_{\frac{3}{2},\frac{3}{2}}$ the binomials become: $$ \binom{1}{n} \binom{2}{n-1} = \frac{1}{n!(1-n)!} \frac{2!}{(n-1)!(3-n)!} $$ which is only possible for the values: $n=1$.
Does the sum run over only at the values of $n$ in which the expression is defined (positive arguments in the factorial)?
Is my representation of the binomials correct? I have seen it throughout many sources.