Color coding to get an FPT algoirthm for k disjoint triangles The k-disjoint triangles problem is as follows:

Input: A graph $G=(V,E)$ and an integer $k\in \mathbb{N}$
Output: Are there $k$ vertex-disjoint triangles in $G$?

An FPT algorithm is presented here
(starting from slide 60). The algorithm uses color-coding and relies on dynamic programming to determine if a solution is highlighted (each vertex in the solution group is colored with a distinct color). The running time of the algorithm is $O^∗((2e)^{3k})$.
I`m trying to understand the running time of this algorithm using dynamic programming (method 2). why is it   $O^∗((2e)^{3k})$?
The dynamic programming algorithm goes over all subsets of $[3k]$ whose size is a multiple of 3, in nondecreasing order of size. For each such non-empty subset, it goes over all $O(n^3)$ triplets of vertices, and for each one, it performs a single table look-up. Therefore the running time is $O^*(2^{3k} n^3)$, where $O^*$ hides $\operatorname{poly}(k)$ factors which are needed to manipulate indices.