On the asymptotic expansion of $\Gamma(x)$, Lagrange's inversion theorem   and the Stirling coefficients
  We show how the asymptotic expansion for the gamma function $\Gamma(x)$, similar to that obtained by Boyd [Proc. Roy. Soc. London A447 (1994) 609--630], can be obtained by using a form of Lagrange's inversion theorem with a remainder. A (possibly) new closed-form representation for the Stirling coefficients is given. 
R. B. Paris
math.CA