Can Inverse z Transform have more than one solution?

Question

I have two inverse of the following z transform:
$$X(z)=\frac3{z-2},|z|>2$$

First Solution

$$\frac{X(z)}z=3\left[\frac1{z(z-2)}\right]$$
By partial fraction, $$\frac{X(z)}z=\frac32\left[\frac1{(z-2)}-\frac1z\right]$$
$$\Rightarrow X(z)=\frac32\left[\frac{z}{(z-2)}-1\right]$$
Clearly it's inverse is:
$$x[n]=\frac32(2^nu[n]-\delta[n])$$

Second Solution

$$X(z)=3z^{-1}\left(\frac{z}{z-2}\right)$$
By using time shifting property,i.e. $x[n-n_0]\leftarrow\rightarrow z^{-n_0}X(z) $,
$$x[n]=3(2)^{n-1}u[n-1]$$

Which one is correct?
Actually both are same except at $n=0$, where I'm wrong?

Answer 1

Ok, my misunderstanding, it's $\delta[n]=1$ , but I thought $\delta[n]=infinity$ at $n=0$