(49g 50g) Fast Pascal's triangle and its relatives

[John Keith]

The programs in the first post have been updated with shorter and faster versions. The first program in the ConvOffs Transform thread has also been updated with the same optimization.

Additional note: The connection between the two programs is that the Narayana triangle is made by transforming the triangular numbers, which are the partial sums of the natural numbers. This idea can be extended ad infinitum by repeated summation.

For example, the tetrahedral numbers are the partial sums of the triangular numbers. The resulting triangle is A056939. This and other examples are shown in Figure 1 of this paper, although the algorithm used in the paper is a different one.

This can also be extended into the realm of transforms. The Narayana transform is related to the binomial transform in the same way that the Narayana triangle is related to Pascal's triangle. Analogous new transforms can be made based on A056939 etc. but as far as I know these theoretical transforms have never been explored.