For those unaware of multifactorial notation, it should be noted that there are some common mistakes made when first being introduced to the notation. For example, ##n!!

eq (n!)!## and ##n!!!

eq (n!!)!

eq (n!)!!

eq ((n!)!)!##. Just to make sure we're all up to speed, here's a quick run down of how multifactorials are defined for integer values of ##n##: $$n!=n(n-1)(n-2)(n-3)...(n-a)|\text{ } (n-a) > 0$$ $$n!!=n(n-2)(n-4)(n-6)...(n-a)|\text{ } 2 \geq (n-a) > 0$$ $$n!!!=n(n-3)(n-6)(n-9)...(n-a)|\text{ } 3 \geq (n-a) > 0$$ $$\large \text{For a more generalized definition:}$$ $$n!^{k}= \left. \begin{cases}

1 & n=0\\

n & 1\leq n\leq k \\

n(n-k)(n-2k)(n-3k)...(n-a) & n>k

\end{cases}\right| k \geq (n-a) > 0$$ Continuing on from this, the next step is to define a function, ##f_k(x)=x!^{k}##, such that ##f_k## is real and continuous for all values of ##x##, which can be accomplished like so:

$$x!^{k}= \left. k^{\frac{x}{k}} \cdot \Gamma(1+\frac{x}{k}) {\prod_{j=1}^{k}}\left(\left(\frac{j \cdot k^{\frac{-j}{k}} }{\Gamma(1+ \frac{j}{k})}\right)^{\displaystyle \frac{1}{k} \sum_{n=1}^{k}\cos\left(\frac{2\pi n\left(x-j\right)}{k}\right)}\right)\right| x \in \mathbb{R}, k \in \mathbb{Z}^+$$ $$\large { \text{Where } !^{k} \text{ represents }k\text{ number of factorial symbols after }} x$$ Now for the second function in this question we need to talk about tetration and subfactorials. Focusing on the latter for now, lets look at the definition for a subfactorial (also known as derangement): $$!n=\text{subfactorial}(n)=\frac{\Gamma(n+1, -1)}{e}$$ $$\large \text{Where } \Gamma(a, x) \text{ is the incomplete gamma function}$$ This leaves us with the final topic of tetration. Tetration is a very useful tool and can be seen as another way of expressing Knuth's up-arrow notation. Consider the following: $$^{n}a=a\uparrow\uparrow n = \Large\underbrace{a^{a^{a^{.^{.^{.^a}}}}}}_{n \text{ times}}$$ Note: the notation can be kind of tricky here; "##n## times" means that the total number of '##a##'s present is equal to ##n##. For example: $$\Large ^{4}3={3^{3^{3^{3}}}}=10^{10^{12.56090264130034}}$$ When this is then applied to create an infinitely tall power tower we end up with this interesting relationship: $$\large ^{\infty}z=z^{z^{z^{.^{.^{.^z}}}}}=h(z)=\frac{-W(-\ln(z))}{\ln(z)}$$ $$\large \text{Where } W(z) \text{ is the Lambert W function}$$ It should be duly noted that ##h(z)## will converge if and only if ##z## falls in the domain ##[e^{-e}, e^{\frac{1} {e}}]##. $$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$$ Now, with the definitions out of the way and hopefully everyone is on the same page, we can finally address the question; $$\large \text{Find the smallest integer value of } k \text{ for which}$$ $$\large f_k(x)=x!^{k} \text{ and }h(!x)=\frac{-W(-\ln(!x))}{\ln(!x)}$$ $$\large \text{have the most points of intersection for } x \geq 0$$ $$\large \text{and evaluate the coordinates for each point, correct to }3 \text{ decimal places}$$