I have calculated the Discrete Time Fourier Transform of the function $x(n)=(1/4)^{n}u(n+4)$ (where $u$ is the Unit Step function) and the result I came up with is:
\begin{equation} X(\omega)=\frac{1024 \cdot e^{4i\omega}}{4-e^{-i\omega}} \end{equation}
Is the result right or wrong?
It looks right to me: $$\sum_{n=-4}^{\infty}\left(\frac{1}{4}\right)^ne^{-i\omega n} = \left(\frac{1}{4}e^{-i\omega}\right)^{-4}\sum_{n=0}^{\infty}\left(\frac{1}{4}\right)^{n}e^{-i\omega n} = (4e^{i\omega})^4\left(\frac{1}{1 - \frac{1}{4}e^{-i\omega}}\right)$$ which is the same as what you wrote, if my mental arithmetic is to be believed.