[borisp]

I left out the signs because they become +1 in the end. The (-1)^n from the -4 and the (-1)^n from the -1/2 join forces to become (-1)^2n = +1.

So there really aren't any sign errors, in that the result is correct. I was just lazy with the typing. Technically the first term should be (-4)^n*(-1/2 choose n).