Given real numbers $x_1,\dots,x_n \in \mathbb{R}$, does there exist a closed form for the expression $$A_n := x_2x_3\cdots x_n + x_1x_3\cdots x_n + \dots + x_1x_2\cdots x_{n-1} = \sum_{i=1}^n \prod_{j=1 \\ i\ne j}^n x_j$$
Obviously the expression can be directly calculated, but is there a form that takes significantly less time to calculate? Maybe with some smart multiplications, divisions and exponentiation.
I found the following relation for the formula:
$$A_n = x_1x_2\cdots x_{n-1} + x_1x_2\cdots x_{n-2}x_n + \dots + x_2x_3\cdots x_n = \\ = x_1x_2\cdots x_{n-1} + x_n(x_1x_2\cdots x_{n-2} + \dots + x_2x_3\cdots x_{n-1}) = \\ = x_1x_2\cdots x_{n-1} + x_nA_{n-1} = \prod_{i=1}^{n-1} x_i + x_nA_{n-1}$$
While it is possible to get the rate of increase of $A_n$ from the above formula, I was already able to get that from the first form; and I wasn't able to manipulate it so that a formula would pop out. I think there should exist a smart way to perhaps factorize a higher degree polynomial into this term and other easy-to-calculate terms.
Any help would be appreciated.
There is this... $$ A_n = \big(x_1x_2\cdots x_n\big)\;\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right) $$ for the case all $x_i \ne 0$.
If one of them is zero, say $x_1 = 0$, then $A_n = x_2x_3\cdots x_n$.
If two or more of them are zero, then $A_n = 0$.