There are two mistakes in your calculations. The first has been pointed out in [Ahmad Bazzi's answer](https://dsp.stackexchange.com/a/54467/4298). However, I would suggest a different expansion of the given function $X(z)$, which only requires the introduction of two unknown constants:

$$X(z)=2.5+\frac{Az^{-1}}{1-1.5z^{-1}}+\frac{Bz^{-1}}{1-0.8z^{-1}}\tag{1}$$

I'm sure you can determine these two constants yourself.

The second mistake is the inverse transform of the basic function

$$G(z)=\frac{1}{1-\alpha z^{-1}}\tag{2}$$

For $|z|>\alpha$ the inverse $\mathcal{Z}$-transform of $(2)$ is

$$g[n]=\alpha^nu[n]\tag{3}$$

where $u[n]$ is the unit step sequence.

For $|z|<\alpha$, the inverse $\mathcal{Z}$-transform of $(2)$ is

$$g[n]=-\alpha^nu[-n-1]\tag{4}$$

From $(2)-(4)$, the inverse $\mathcal{Z}$-transform of $(1)$ is

$$x[n]=2.5\delta[n]-A(1.5)^{n-1}u[-n]+B(0.8)^{n-1}u[n-1]\tag{5}$$

For $n=0$ you have contributions from two terms:

$$x[0]=2.5-A(1.5)^{-1}\tag{6}$$