  We obtain an explicit simple formula for the coefficients of the asymptotic
expansion for the factorial of a natural number,in terms of derivatives of
powers of an elementary function. The unique explicit expression for the
coefficients that appears to be known is that in the book by L. Comtet, which
is given in terms of sums of associated Stirling numbers of the first kind. By
considering the bivariate generating function of the associated Stirling
numbers of the second kind, another expression for the coefficients in terms of
them follows also from our analysis. Comparison with Comtet's expression yields
combinatorial identities between associated Stirling numbers of first and
second kind. It suggests by analogy another possible formula for the
coefficients, in terms of a function involving the logarithm, that in fact
proves to be true. The resulting coefficients, as well as the first ones are
identified via the Lagrange inversion formula as the odd coefficients of the
inverse of a pair of formal series, which permits us to obtain also some
recurrences.
