In the OEIS entry for Bell numbers, there appears a generating function
$$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$
However, I could not locate any proof of reference for this formula. The contributor informs me that he discovered it by experimentation.
I would appreciate any further information on this generating function.
Best Answer
The proof is given, for example, in http://www.sciencedirect.com/science/article/pii/S0097316503000141 (Bell numbers, their relatives, and algebraic differential equations, by Martin Klazar). Namely it is proved that the generating function $B(t)=\sum\limits_{n=0}^\infty B_nt^n$ satisfies the functional equation $$B(t)=1+\frac{t}{1-t}B\left(\frac{t}{1-t}\right).$$ Iterating this equation, we get (Klazar calls it the classical expansion of B(t)) $$B(t)=\sum\limits_{n=0}^\infty \frac{t^n}{(1-t)(1-2t)\cdots(1-nt)}.$$