Weighted sum of specific multinomial coefficients

Question

Let $A$ and $b$ be nonnegative integers and consider the sums $$\sum\limits_{c=0}^{b/2}\frac{1}{4^c}\binom{A}{c,b-2c,A-b+c}$$ and $$\sum\limits_{c=0}^{b/2}\frac{c}{4^c}\binom{A}{c,b-2c,A-b+c}.$$ I would like closed forms for each of these; Mathematica gives an answer, for instance the latter is supposedly $$\frac{A(-2A+b-1)!}{2^b(1-2A)!(b-2)!}$$ but this is nonsensical according to the Mathematica documentation since the factorial of a negative number (such as $1-2A$) is always ComplexInfinity. So, does anyone know the correct way to go about evaluating these sums, or what Mathematica is trying to do?