Same z transformed function, but different answers of inverse z transform?

Given z transformed function is E(z)=1/(z+4)

I know there are several ways to get the inverse z transform of this function.

1. use partial fraction

E(z)=1/(z+4)

E(z)/z=1/z(z+4) apply partial fraction here,

E(z)/z=(1/4)*((1/z)-(1/(z+4)) so E(z) is,

E(z)=(1/4)*(1-(z/(z+4)) as you know, it is easy to use inverse z transform here.

e(k)=(1/4)*(delta(k)-(-4)^k)

1. inversion formula method

E(z) has simple pole at z=-4, the residue is evaluated as

{(residue)|z=-4} = {(z+4)*E(z)*z^(k-1)|z=-4

             = {1*z^(k-1)|z=-4}

= (-4)^(k-1)

= -(1/4)*(-4)^k


The problem is that inverse z transform of the same function E(z)

has 2 different answers. As you see, there is no delta function when I used

inversion formula method.

At first, I thought it was because a simple pole,z=-4, is outside of the unit

circle, but same things happen when I used a simple which is inside of the unit circle.

Why does this difference happen?

The equations do produce the same results. $\small e(k)$ is assumed unilateral, so the equations in method 2 should be qualified by $\small k>0$, thus:
$$\small e(k)=\begin{cases} (-4)^{k-1}= -(1/4)(-4)^{k}, \:\:\:k>0 & \\\\ 0, \:\:\text{otherwise}\end{cases}$$
$$\small \delta (k)=\begin{cases} 1, \:\:\:k=0 & \\\\ 0, \:\:\text{otherwise}\end{cases}$$