How to interpret Pascal's Triangle for negative numbers?

[PLAGUE]

TL;DR

Intuition behind extended Pascal's Triangle.

This answer shows an extended version of Pascal's Triangle that works for negative numbers too.

In This video, Sal shows how to interpret the members of Pascal's Triangle as the sum of all the possible paths to get to that member.

Is there any way we can use this same 'sum of all the possible paths' to interpret This extended version of Pascal's Triangle?

 

[pbuk]

In This video, Sal shows how to interpret the members of Pascal's Triangle as the sum of all the possible paths to get to that member.

I'm not going to watch a video, but the interpretation as the sum of all possible paths follows directly from the interpretation as the sum of two entries in the preceding row: think about it.

Is there any way we can use this same 'sum of all the possible paths' to interpret This extended version of Pascal's Triangle?

Can you see that the magnitude of the numbers in the 'negative Pascal's triangle' are the same as those in the normal version? Can you work out how the rearrangement works? Can you work out how the signs are allocated?

Once you have done this the answer should be clear, not just to your question but to how any fact about the normal Pascal's triangle translates to the negative one.

 

[PLAGUE]

I think I know the answers to your questions.

But is there any physical significance at all?

 

[pbuk]

But is there any physical significance at all?

IMHO Pascal's triangle has no 'physical significance', it is just a diagrammatic representation of a recurrence relation. It is interesting that the diagram displays more patterns than one might expect, but these result from the underlying recurrence relation not from the diagram.