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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.

1 121

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

1 121 12321

And 1 - 2- 3 - 4 on this?

1 121 12321 1234321

Can you generalize?

1 121 ... 12345654321

Can you explain what you find?