  This paper is the first application of the compensation approach to counting
problems. We discuss how this method can be applied to a general class of walks
in the quarter plane $Z_{+}^{2}$ with a step set that is a subset of
$\{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}$ in the interior of $Z_{+}^{2}$. We
derive an explicit expression for the counting generating function, which turns
out to be meromorphic and nonholonomic, can be easily inverted, and can be used
to obtain asymptotic expressions for the counting coefficients.
