Unit impulse response $h[n]$ of a discrete time system with multiple characteristic roots 0
If you have a difference equation
$$\sum_{k=0}^Na_ky[n-k]=\sum_{k=0}^Mb_kx[n-k]\tag{1}$$
and if $M&gt;N$, the general expression for the corresponding causal impulse response is
$$h[n]=A_0\delta[n]+A_1\delta[n-1]+\ldots+A_{M-N}\delta[n-(M-N)]+u[n]\sum_{k}c_k\gamma_k^n\tag{2}$$
where $\gamma_k$ are the roots of the characteristic equation, and where I've assumed simple roots. Note that roots $\gamma_k=0$ do not contribute to $(2)$ (even if there are multiple roots equal to zero).
For the given example we have $M-N=3$ and one non-zero root $\gamma=\frac12$. Hence the solution has the form
$$h[n]=A_0\delta[n]+A_1\delta[n-1]+A_2\delta[n-2]+A_3\delta[n-3]+c\left(\frac12\right)^n\tag{3}$$
The constants $A_i$, $i=0,1,\ldots,3$, and $c$ must be determined by iterating through the difference equation with $x[n]=\delta[n]$, $h[n]=y[n]$, and $h[-1]=0$.
If you do that you'll find that $c=0$, which means that the given difference equation describes an FIR system. In terms of transfer functions this means that there is a pole-zero cancellation, leaving only poles at the origin of the complex plane.