Challenge Level

There are three ways to enter the triangular array of numbers in the diagram below, starting with 1, and counting from 1 -2 in order, finishing with the two in the bottom row.

How many ways are there to count 1 - 2 - 3 in the following array?

And 1 - 2- 3 - 4 on this?

Can you generalize?

Can you explain what you find?