In the [OEIS entry](https://oeis.org/A000110) 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.



Ordinary Generating Function for Bell Numbers
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))
$$B(t)=\sum\limits\_{n=0}^\infty \frac{t^n}{(1-t)(1-2t)\cdots(1-nt)}.$$


