  A conjectured relation between Ramanujan's asymptotic approximations to the
exponential function and the exponential integral is established. The proof
involves Stirling numbers, second-order Eulerian numbers, modifications of both
of these, and Stirling's approximation to the gamma function. Our work provides
new information about the coefficients in Stirling's approximation and their
connection to Ramanujan's approximation coefficients. A more analytic second
proof of the main result is also included in an appendix.
