Angle of a regular simplex I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.

What is the vertex angle of a regular $n$-simplex?

Background: For a vertex $v$ in a convex polyhedron $P$, the vertex angle at $v$ is the proportion of the volume that $P$ occupies in a small ball around $v$. In symbols, $$\angle v=\lim_{\varepsilon\to 0} \frac{|B(v,\varepsilon)\cap P|}{|B(v,\varepsilon)|}.$$ Up to normalization, this definition agrees with the familiar definition of the angles in the plane, or the solid angle in $3$-space.
 A: In Theorem 1 in this paper (also citation 3 in that paper), a different definition of the vertex angle is given, with the value for the regular simplex computed. If you combine that with the description in Mathworld, you should be able to get an expression for your expression up there. 
A: Here is a link to the original paper by Rogers mentioned by Joseph O'Rourke in his answer, with a short proof of the asymptotic formula for $F_n(\alpha)$, which is reproduced in the book Sphere Packings by Chuanming Zong.
It is also shown there that the multiplicative error in the formula is at most $1+ \frac{31}{12n} + O\left(\frac{1}{n^2}\right)$ in the case of the regular simplex, where $\sec 2\alpha=n$ (and thus $c=1$, or, $b=1$ in the paper).
A: In the paper by John Leech,
"Sphere packings in Higher Space"
Canadian Journal of Mathematics, 1964, which you can find at the Google book links here,
the following formulas are given for the "solid angular content at each vertex of a regular simplex":
$$2^{-n} n! f_n(n) H_n$$
where
$$H_n = 2 \pi^{\frac{1}{2}n} / \Gamma(n/2)$$
is the total $(n{-}1)$-dimensional surface content,
$f_n(\sec 2 \alpha)=F_n(\alpha)$, and (finally!)
$F_n(\alpha)$ is Schläfli's function mentioned by BS in his comment.
This function is discussed in Section 7.2 (p.107ff) of
Chuanming Zong's book, Sphere Packings (Google books link here)
and in
the paper, "Analytic structure of Schläfli function,"
Kazuhiko Aomoto,
Nagoya Math. J. Volume 68 (1977), 1-16,
also mentioned by BS.
Rogers computed an asymptotic formula for $F_n(\alpha)$:
$$ \frac{ \sqrt{ 1+cn}} {\sqrt{2} e^{1/c} n!} \left( \frac{2 e}{\pi c n} \right) ^{n/2} \;,$$
where
$c = (\sec 2 \alpha - (n-1))^{-1}$.
