Is there a way to generalize the polygonal and pyramidal numbers to higher dimensions?

Question

I know that there exists formulae for the polygonal numbers and pyramidal numbers, as stated on OEIS here. So far, I haven't been able to find anything anywhere related to a generalization of these numbers to higher dimensions, however. I was able to find and prove a result by induction for the generalization of triangular and tetrahedral numbers to higher dimensions, namely $$\frac{(x(x+1)(x+2)…(x+n-1))}{n!}$$ where $x$ is the $x^{th}$ term for a given dimension $n$. This apparently correspons to the simplicial polytopic numbers, as noted on both OEIS and Wikipedia. However, this is only related to the n-dimensional simplex. Any generalization of the polygonal and pyramidal numbers as a whole to higher dimensions has eluded me, and nothing on the internet seems to help.
I'm currently a high school student, so I would appreciate if answers can be kept elementary if possible. If not, then I'll just do my best to understand.

Answer 1

Congratulations on deriving the $n$-simplicial numbers!

Personally, I find the simplicial numbers (and of course the $n$-cubic numbers, that is, the powers of $x$) to be a lot more useful than the other polygonal numbers. The arrangements of dots to form a pentagonal number seem "off" to me (not nearly as much symmetry as the triangular and square numbers can attain) and I'm not sure what you would do with, say, dodecahedral numbers.

A "$600$-cell number" in four dimensions seems even weirder. Visualizing the $600$-cell polytope itself is hard enough without having to arrange dots in it.

Beyond four dimensions, there aren't any regular polytopes except the simplexes, the $n$-cubes, and the orthoplexes (duals of the $n$-cubes), so the opportunities for higher-dimensional figurate numbers seem somewhat limited.

We have a bit more luck with square pyramids; they will have higher-dimensional analogues in all dimensions because the square and triangle do. And the square-pyramidal numbers in higher dimensions will be sums of consecutive $n$th powers, which is something interesting.

You might also look at the orthoplexes--the two-dimensional orthoplex is the same as the square, but the three-dimensional one is the octahedron. The $(n-1)$-faces of an $n$-orthoplex will all be $(n-1)$-simplexes.

The OEIS does feature a number of other figurate sequences besides the simplicial numbers and the polygonal numbers; the entry for the octahedral numbers links to several other examples.