TITLE: Closed form expression for a recursion relation with binomial coefficients
QUESTION [6 upvotes]: I am interested in the following sequence: $$ T_n = \sum\limits^{n-1}_{k=0} \begin{pmatrix} n \\ k \end{pmatrix} T_{k}, \ \ \ \ T_0 = C \in \mathbb{N} $$
I would like to express it as a function of n, but none of the method I have tried work.
Asymptotically, I can tell that $T_n = \mathcal{O}(2^{\frac{k^2}{2}})$.
One method that failed was to see $T_n$ as the $n$-th term in a series, but those terms grow to fast for it to work.
Do you know how to solve it, or have an intuition regarding how it might get solved?
Thank you.

REPLY [9 votes]: The $T_n$'s are equal to the product of $C$ and the Fubini numbers: number of ordered partitions of $n$, also known as ordered Bell numbers. The generating function is $(2-e^x)^{-1}$ and the large-$n$ asymptotics is 
$$T_n\sim C \frac{n!}{2(\ln 2)^{n+1}}.$$