derivation of recurrence for Sylvester’s sequence

Let us begin with the product:

$$a_n=1+\prod _{i=0}^{n-1}{a}_i$$

Adding $1$ to $n$ and manipulating the result:

$a_{n+1}$$=$$1+{\displaystyle \prod _{i=0}^n}{a}_i$

$=$$1+a_n{\displaystyle \prod _{i=0}^{n-1}}{a}_i$

$=$$1+a_n(a_n-1)=1+{(a_n)}^2-a_n$