Is the formula $\sum _{m=1}^{n-1} \prod _{k=m+1}^n x_k x_m$ wrong in the wiki's page

Question

SymmetricPolynomial[2, {Subscript[x, 1], Subscript[x, 2], Subscript[x,   3], Subscript[x, 4]}]

$$\begin{align*}x_1 x_2+x_3 x_2+x_4 x_2+x_1 x_3+x_1 x_4+x_3 x_4\tag{1}\end{align*}$$

Plus @@ Subsets[  Times[Subscript[x, 1], Subscript[x, 2], Subscript[x, 3], Subscript[   x, 4]], {2}]

$$\begin{align*}x_1 x_2+x_3 x_2+x_4 x_2+x_1 x_3+x_1 x_4+x_3 x_4\tag{2}\end{align*}$$

However if I follow the formula (3) from this page.

Then, I'll not get (1), but get (4)

$$\begin{align*}\sum _{m=1}^{n-1} \prod _{k=m+1}^n x_k x_m\tag{3}\end{align*}$$

\!\(\*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(n - 1\)]\(\*UnderoverscriptBox[\(\[Product]\), \(k = m + 1\), \(n\)]\*SubscriptBox[\(x\), \(k\)]\ \*SubscriptBox[\(x\), \(m\)]\)\) /. n -> 4

$$\begin{align*}x_2 x_3 x_4 x_1^3+x_2^2 x_3 x_4+x_3 x_4\tag{4}\end{align*}$$

Since $\prod _{k=m+1}^n x_kx_m$ may be some notations that I'm not familiar with, I'm not sure whether the formulas in the wiki's page is wrong.

---

Replace product in (3) with summation, then we will get the right/expected result.

\!\(\*UnderoverscriptBox[\(\[Sum]\), \(m = 1\), \(n - 1\)]\(\*UnderoverscriptBox[\(\[Sum]\), \(k = m + 1\), \(n\)]\*SubscriptBox[\(x\), \(k\)]\ \*SubscriptBox[\(x\), \(m\)]\)\) /. n -> 4

$$\begin{align*}x_1 x_2+x_3 x_2+x_4 x_2+x_1 x_3+x_1 x_4+x_3 x_4\tag{5}\end{align*}$$

Are the formulas wrong? or Is my calculating method in Mathematica wrong?

Note that some notations with the complex use of $\Pi$ and $\sum$ in symmetric polynomials

Answer 1

[This answer was made while the OP question kept being changed...I hope it is current.]

The easy way is to use the built-in SymmetricPolynomial:

vietePoly[deg_Integer, n_Integer, var_: \[FormalX]] := 
 SymmetricPolynomial[deg, Array[var, n]]

Clear[x];
vietePoly[2, 4, x]
(* x[1] x[2] + x[1] x[3] + x[2] x[3] + x[1] x[4] + x[2] x[4] + x[3] x[4] *)

The formula for degree 2 should have a sum instead of product:

$$\sum _{m=1}^{n-1} \sum _{k=m+1}^n x_k x_m $$

The general formula for the symmetric polynomial of degree $k$ is

$$s_k = \sum_{1 \le i_1 < i_2 < \cdots < i_k \le n} x_{i_1} x_{i_2} \cdots x_{i_k}$$

Sum[x[m] x[k], {m, 1, n - 1}, {k, m + 1, n}] /. n -> 4 // Expand
(* x[1] x[2] + x[1] x[3] + x[2] x[3] + x[1] x[4] + x[2] x[4] + x[3] x[4] *)