I noticed and employed (without a problem) an approximation for Stirling's number of the second kind found on Wikipedia (http://en.wikipedia.org/wiki/Stirling_numbers_of_the_second_kind), in particular this approximate expression: (http://upload.wikimedia.org/math/8/a/f/8afeed15dd40295320cf974418895cc2.png):

$n \brace k$$\space \approx \frac{\sqrt{n-k}}{\sqrt{n(1-G)}G^k(v-G)^{n-k}} (\frac{n-k}{e})^{n-k}$ ${n}\choose{k}$$,\space \space \space \forall{k}$ s.t. $1<k<n$

Where:

$G = -W_0(-ve^{-v}) \space \space \space$ ($W_0$ being the main branche of Lambert W function)

$v = (\frac{n}{k})$

However, upon looking up the stated references [Ref. 13 and 14 on the Wikipedia page]:

13. W. E. Bleick and Peter C. C. Wang, Asymptotics of Stirling Numbers of the Second Kind, AMS Vol.42 No.2, 1974.

14. N. M. Temme, Asymptotic Estimates of Stirling Numbers, STUDIES IN APPLIED MATHEMATICS 89:233-243 (1993), Elsevier Science Publishing.

It's not clear to me where this exact expression comes from (though approximate expressions in the same vein are presented in both papers that rely on the Lambert W function). Can anyone help me out in terms of understanding where the exact form of the expression on Wikipedia comes from and/or what liberties are being taken by the Wikipedia author of the expression?

For convenience, here's the expression (http://upload.wikimedia.org/math/8/a/f/8afeed15dd40295320cf974418895cc2.png) in Mathematica format:

v = n/k;

G = -ProductLog[0, -v*E^(-v)];

stirlingApprox = (n - k)^(1/2)/((n*(1 - G))^(1/2)G^k(v - G)^(n - k))*((n - k)/E)^(n - k)*Binomial[n, k];

Note that "StirlingS2[n, k]" directly gives the Stirling number of the second kind in Mathematica.

The approximation appears to satisfy the stated relevant error of $\approx (\frac{0.06}{n})$, however I haven't checked this carefully yet.