I think you're confusing the value of the function at with the point at which the power series is centered. Since both the series for and are centered at , by using the product of the power series you're already expanding around . Also notice that the Laurent series is just the Taylor expansion since it's a product of entire functions.

Now using the Cauchy product formula we have:

Let and then where

So to compute the second term in the expansion we have which would be the second coefficient in the series (if I made no mistake), now just compute two more which are non-zero.