\documentclass[english]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\usepackage{babel}
\begin{document}
% ...
\[
\begin{array}{ccccc}
x(1) & = & \frac{x(0)}{1} & = & x(0)\\
x(2) & = & \frac{x(1)}{2} & = & \frac{x(0)}{2}\\
x(3) & = & \frac{x(2)}{3} & = & \frac{x(0)}{2.3}\\
x(4) & = & \frac{x(3)}{4} & = & \frac{x(0)}{4!}\\
& & & & \vdots\\
x(n) & = & \frac{x(n-1)}{n} & = & \frac{x(0)}{n!}
\end{array}
\]
% ...
\end{document}
Because of the fractions in the array cells, array elements are vertically too close to each other, they are literally touching to each other. Also the horizontal spacing is too much as you see in the image (screen shot from the output PDF file).
How do I make these cell-spacings look better?
\\
, i.e.\\[2pt]
and\\[-2pt]
resp. add/remove 2pt of vertical space.x(n-1)/n
in the middle column, notx(n-1)/n!