Yo dawg, I heard you like partial derivatives

Let a function $f$ in three variables $x,y,z$ be defined as follows :

$f(x,y,z)=\ln(x^3+y^3+z^3-3xyz)$

Also, we define the following :

$\LARGE \mathcal{E}_n(x,y,z)=\sum_{i\in\{x,y,z\}}f_{\underbrace{iii\ldots ii}_{n\textrm{ times}}}~\forall~n\in\Bbb{Z^+}$

If the value of $\mathcal{E}_{101}(1,0,0)$ can be represented as $A\cdot B!$ for integers $A,B$ where $A$ is a prime, then submit your answer as the value of $(A+B)$ .

Bonus: Find the explicit formula(s) for $\mathcal{E}_n(x,y,z)$ where $n\in\Bbb{Z^+}$ .

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This problem is inspired by one of my calculus professors.

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Notations used :

• For a function $f$ of $n$ variables $x_1,x_2,\ldots,x_n$ , we denote $\dfrac{\partial f}{\partial x_i}$ as the partial derivative of $f$ w.r.t $x_i$ where $i\in\{1,2,\ldots,n\}$ .

• $f_{a_1a_2\ldots a_n}=\dfrac{\partial}{\partial a_1}\left(\dfrac{\partial}{\partial a_2}\left(\cdots\left(\dfrac{\partial f}{\partial a_n}\right)\cdots\right)\right)$ where $f$ is a function of $n$ variables $a_1,a_2,\ldots,a_n$ .

• $\ln(\cdot)$ denotes the natural logarithm .