# Tetrahedron trinomial coefficient transform

## TL;DR

This paper introduces the tetrahedron trinomial coefficient transform, a new mathematical operation on Pascal-like structures that reveals relationships between different edges of an infinite tetrahedral array, generalizing binomial transforms.

## Contribution

It defines a novel tetrahedron trinomial coefficient transform and proves its relation to the structure of an infinite Pascal-like tetrahedron, extending binomial transform concepts.

## Key findings

- The transform maps a face of the tetrahedron to its opposite edge.
- Application to Pascal's triangle yields the binomial transform of central binomial coefficients.
- The method generalizes binomial transforms to three-dimensional structures.

## Abstract

We introduce the tetrahedron trinomial coefficient transform which takes a Pascal-like arithmetical triangle to a sequence. We define a Pascal-like infinite tetrahedron H, and prove that the application of the tetrahedron trinomial transform to one face T of H provides the opposite edge E to T in H. It follows from the construction that the other directions in H parallel to E can be obtained similarly. In case of Pascal's triangle the sequence generated by the trinomial transform coincides the binomial transform of the central binomial coefficients.