The Five Variable Triangular Dipyramid is constructable based on a permutation of the Octahedron so that it is triangular rather than square (vertically).

The summation requires a checkerboard of numbers, omitting half of them from relevancy to the Pascalloid.

and here is the cominatoric expression for a Triangular Dipyramid:

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This interesting calculation produces results that form the geometric dual of the prism, five corners and six sides instead of the opposite. However unlike prisms, these don't tile through space.

And newly discovered is a perfect algebraic equation with terms that have coefficients to match these numbers.

51 = 5
52 = 25
53 = 125
54 = 625
55 = 3,125
56 = 15,625
57 = 78,125