Can you provide a proof for the following claim:

$$-\displaystyle\sum_{n=1}^{\infty}\frac{J_k(n)}{n} \cdot \ln\left(1-x^n\right)=\frac{x \cdot A_{k-1}(x)}{(1-x)^k} \quad \text{for} \quad |x| < 1 \quad \text{and} \quad k>1$$ where $J_k(n)$ is the Jordan's totient function and $A_k(x)$ is the kth Eulerian polynomial.

The first few Eulerian polynomials are: $A_1(x)=1 , A_2(x)=x+1 , A_3(x)=x^2+4x+1$ , etc.

The SageMath cell that demonstrates this claim can be found here.