Today I've come across Michael Stifel's version of Pascal's Triangle, which I've seen referred to as the Figurate Triangle or the Triangle of Figurate Numbers as seen in Combinatorics: Ancient and Modern (page 72) and Between Certainty and Uncertainty (page 93)

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In class we were encouraged to attempt to figure out why his triangle was written the way it was.

Through the research I've come up with, it appears that each column appears to be, in order, the natural numbers, the triangle numbers, the tetrahedral numbers, and so on.

But he is skipping a few of the initial numbers for each column. For example he is missing the triangle number $1$, the tetrahedral numbers $1$ and $4$, etc.

Is there any logical or known reasons why this is the case?

  • 3
    $\begingroup$ It looks like he is saving space. He skipped the column of 1's, and the missing numbers in each row would simply repeat the ones shown in the reverse order. $\endgroup$ – Conifold Mar 21 '19 at 9:12

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