The following sequence appeared in IMC 2012 (a math competition): $$a_1 = \frac{1}{2}, \qquad a_{n+1} = \frac{n a_n^2}{1+(n+1)a_n}$$
I am trying to find an explicit formula for the sequence. It seems to be nicer to look at $b_n = a_n^{-1}$: $$b_1 = 2, \qquad b_{n+1} = \frac{b_n(b_n +n +1)}{n}$$ Can some closed formula be derived?