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We obtain an explicit simple formula for the coefficients of the asymptoticexpansion for the factorial of a natural number,in terms of derivatives ofpowers of an elementary function. The unique explicit expression for thecoefficients that appears to be known is that in the book by L. Comtet, whichis given in terms of sums of associated Stirling numbers of the first kind. Byconsidering the bivariate generating function of the associated Stirlingnumbers of the second kind, another expression for the coefficients in terms ofthem follows also from our analysis. Comparison with Comtet's expression yieldscombinatorial identities between associated Stirling numbers of first andsecond kind. It suggests by analogy another possible formula for thecoefficients, in terms of a function involving the logarithm, that in factproves to be true. The resulting coefficients, as well as the first ones areidentified via the Lagrange inversion formula as the odd coefficients of theinverse of a pair of formal series, which permits us to obtain also somerecurrences.

The asymptotic expansion for the factorial and Lagrange inversion formula