Centered Platonic Numbers

On the 19th November 2018, I created a post titled Platonic Numbers and at the end of that post I made reference to Centered Platonic Numbers as follows:

Additionally, there are the centered Platonic numbers defined by starting with 1 central dot (for \(n\)=0) and adding regular convex polyhedral layers around the central dot, where the \(n\)th layer, \(n\) ≥ 1, has \(n\)+1 dots per facet ridge (face edge for polyhedrons) including both end vertices. The formulae are very similar to the above and can be explored further here.

Let's look a particular type of centered Platonic number, namely the centered tetrahedral numbers, and the case where n=2. See Figure 1.

Figure 1

This second centered tetrahedral number differs from the equivalent tetrahedral number by 1. The progression of the tetrahedral numbers from n=1 to 4 is shown in Figure 2.

Figure 2

With the tetrahedral numbers, there is a layering process in place as is clearly visible in the progression shown in Figure 2. With the centered tetrahedral numbers, the tetrahedra shown in Figure 2 are being built around the central dot. Each progressive tetrahedron contains all the previous tetrahedra, Russian doll style. 

Thus the centered tetrahedral numbers can be derived by progressively summing the tetrahedral numbers:

1, 1 + 4, 1 + 4 + 10, 1 + 4 + 10 + 20 --> 1, 5, 15, 35 etc.

The formulae for the various centered Platonic numbers can be found at this site and for the centered tetrahedral numbers, the formula is:$$ \frac{ (2n+1)(n^2+n+3)}{3}$$This formula generates the following initial members of OEIS A005894:

1, 5, 15, 35, 69, 121, 195, 295, 425, 589, 791, 1035, 1325, 1665, 2059, 2511, 3025, 3605, 4255, 4979, 5781, 6665, 7635, 8695, 9849, 11101, 12455, 13915, 15485, 17169, 18971, 20895, 22945, 25125, 27439, 29891, 32485, 35225, 38115

The generating function for this series is: $$ \frac{(1+x)(1+x^2)}{(1-x)^4}$$For example, the first few terms of the resulting Taylor series are:$$1+5x+15x^2+35x^3+69x^4+121x^5+195x^6+295x^7 + ...$$