How Can We Represent and Sum the Rows of This Manipulated Pascal's Triangle?

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Discussion Overview

The discussion revolves around a manipulated version of Pascal's Triangle and seeks to explore ways to represent the rows in terms of combinations. Participants are also interested in identifying patterns related to the sums of these rows, particularly in the context of a problem involving movement on a grid.

Discussion Character

  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant presents a manipulated Pascal's Triangle and seeks a representation of its rows in terms of combinations.
  • Another participant questions the purpose of the manipulation and suggests a connection to the binomial coefficient squared, (nCk)².
  • A further inquiry is made regarding patterns in the sums of the rows, linking it to a problem about two players moving on a grid and the expected value of their meetings.
  • A participant references an external source that may contain relevant information about the sums of the rows.

Areas of Agreement / Disagreement

The discussion includes multiple viewpoints and inquiries, with no consensus reached on the representation or patterns in the sums of the rows.

Contextual Notes

The exploration is limited by the lack of detailed assumptions regarding the manipulation of Pascal's Triangle and the specific mathematical properties being investigated.

rbzima
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Here is the manipulation of Pascals Triangle. I'm trying to figure out a way to represent these rows in terms of a combination. Check it out...



1
1 1
1 4 1
1 9 9 1
1 16 36 16 1
1 25 100 100 25 1
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.
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Why have you posed this?

(nCk)2
 
mathman said:
Why have you posed this?

(nCk)2

Is there any particular pattern also with the sum of the rows as well?

The reason is because it's part of a problem I'm working on regarding two people at opposite corners on a massive n by n board. On player can only move north and east, the other south and west. Basically, I need to find the expected value of the number of times they will meet, and I have a hunch this plays a part in. Thanks!
 
rbzima said:
Is there any particular pattern also with the sum of the rows as well?

See http://www.research.att.com/~njas/sequences/A000984" .
 
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