  This paper gives an explicit formula of the asymptotic expansion of the
Kobayashi-Royden metric on the punctured sphere
$\mathbb{CP}^1\backslash\{0,1,\infty\}$ in terms of the exponential Bell
polynomials. We prove a local quantitative version of the Little Picard's
theorem as an application of the asymptotic expansion. Meanwhile, the approach
in the paper leads to the conclusion that the coefficients in the asymptotic
expansion are rational numbers. Furthermore, the explicit metric formula and
the conclusion regarding the coefficients apply to a more general case as well,
the metric on
$\mathbb{CP}^1\backslash\{0,\frac{1}{3},-\frac{1}{6}\pm\frac{\sqrt{3}}{6}i\}$
will be given as a concrete example of our results.
