3D analogue of A124278

[Jonathan Post]

What is the 3D analogue of A124278?  We might describe it as:
"Triangle of the number of nondegenerate k-polyhedra having total
edge-length n and whose edge lengths are nondecreasing on each face."
or equivalently: "Triangle of the "number of nondegenerate k-faced
polyhedra having total edge-length n and whose faces are polygons with
nondecreasing sides."

A completely different 3D analogue of A124278 would be: "Triangle of
the number of nondegenerate k-polyhedra having total area n."

Proper phrasing of these definition, and corresponding formulae, would
generalize first to 4D analogues, and then higher dimensions.