B How to interpret Pascal's Triangle for negative numbers?

AI Thread Summary
An extended version of Pascal's Triangle can be interpreted for negative numbers, maintaining the same magnitudes as the standard triangle. The interpretation as the sum of all possible paths to a member is applicable to this extended version, following the same logic as the standard triangle. Understanding how the signs are allocated and how the rearrangement works is crucial for clarity on this topic. While some participants question the physical significance of Pascal's Triangle, it is primarily viewed as a representation of a recurrence relation rather than a model with physical implications. Overall, the discussion emphasizes the mathematical relationships and patterns inherent in both the standard and negative versions of Pascal's Triangle.
PLAGUE
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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?
 
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PLAGUE said:
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.

PLAGUE said:
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.
 
I think I know the answers to your questions.

But is there any physical significance at all?
 
PLAGUE said:
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.
 
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