Is there an explicit formula for power series of $\left(\frac{1+x}{1-x}\right)^n$?

Is there an explicit formula for power series of
$\left(\frac{1+x}{1-x}\right)^n$?

I am trying to answer this question. Per suggestion in one the comments
for that question, one might be able use the power series of the terms to
arrive at the answer. However, one of the terms is:
$$\left(\frac{1+x}{1-x}\right)^n$$
I am wondering if there is an explicit power series formula for this
expression when $x$ is small.
I cannot see a pattern in the expression for the Taylor series output by
Mathematica:
$\left(\frac{1+x}{1-x}\right)^n=1+2 n x+2 n^2 x^2+\frac{2}{3} \left(2
n^3+n\right) x^3+\frac{2}{3} \left(n^4+2 n^2\right) x^4+\frac{2}{15}
\left(2 n^5+10 n^3+3 n\right) x^5+\frac{2}{45} \left(2 n^6+20 n^4+23
n^2\right) x^6+\frac{2}{315} \left(4 n^7+70 n^5+196 n^3+45 n\right)
x^7+\ldots$