Presumably this is a well-known identity. Does it have a name?pic.twitter.com/tsMBGLZGAk
You can add location information to your Tweets, such as your city or precise location, from the web and via third-party applications. You always have the option to delete your Tweet location history. Learn more
So these S(n, k) are just the entries of the matrix inverse to that whose entries are the coefficients s(n, k) appearing in the expression for x^n in terms of the falling-factorial polynomials.
And that is how I am accustomed to think of the Stirling numbers of the two different kinds -- that is, I am accustomed to regard them as *defined* by their respective roles as matrix elements in these two (mutually inverse) basis-change matrices.
Twitter may be over capacity or experiencing a momentary hiccup. Try again or visit Twitter Status for more information.