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]:
W. E. Bleick and Peter C. C. Wang, Asymptotics of Stirling Numbers of the Second Kind, AMS Vol.42 No.2, 1974.
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.
As an update --- "The approximation appears to satisfy the stated relevant error of $\approx (\frac{0.06}{n})$, however I haven't checked this carefully yet."
I've now checked $n \brace k$ for $n \in [2, 1000]$ and $\forall k < n$ for each $n$. I found: (1) that the value for $\delta = |(1 - \frac{(exact)}{(approximation)})|$ decreases monotonically with increasing $n$ (regardless of the value of $k < n$); and (2) that for each value of $n$, the value of $\delta$ depends on $k$ in a particular, though consistent manner for most values of $n$. More specifically, there seems to be one local maxima for the smaller values of $k$, a global maxima appears near the mid-range values of $1<k<n$, and then we see that $\delta$ decreases monotonically after the global maxima with increasing $k$.
The largest value of the error we encounter is $\delta_{max} = 0.040932861725805439857312709933506101086741032506081...$ for $(n,k) = (2,1)$.
Now, in terms of whether $\approx (\frac{0.06}{n})$ is a good upper bound for $\delta$, actually it's not conservative enough -- we can notice that all values of $\delta$ very slightly exceed this bound. So perhaps $\approx (\frac{0.066}{n})$ is a more appropriate upperbound, as it is satisfied by all values of $\delta$ for $n \geq 6$.