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Topic: Shapes of binomial #s
Replies: 3   Last Post: Jan 20, 1995 5:28 AM

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Posts: 14
Registered: 12/6/04
Shapes of binomial #s
Posted: Jan 19, 1995 3:24 PM
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This is a conjecture about the shapes of numbers that are coefficients of
the different terms in (a+b)^n

Below I've attempted to draw the first four coefficients of (a+b)^6
(which are 1, 6, 15, and 20)

1 1 1 1 1
1 1
1 11 11 11
1 111 111 111
1 1111 1111
1 11111

As is well known, the third coefficient is a triangle number. Looking at
these numbers I believe the first coeeficient could be described as a
point number and the second as a line number.

The fourth coefficient is the sum of consecutive triangle numbers. If
these triangles are stacked on top of each other they form a tetrahedron.
(I have a more clear rendition of this in the JPEG format. If you would
like to see it, let me know via e-mail)

To me this suggests a sequence:

The first coefficient of a binomial expansion is a point number
It is one point.

The second coefficient is a line number
Lines are bound by 2 points

The third is a triangle number
It is bound by 3 lines.

The fourth, a tetrahedron number.
Tetrahedrons are bound by 4 triangles.

To me this suggests a sequence. Would the next coefficient be a number
that forms an object in 4-space? Presumably this object would be bound by
5 tetrahedrons. The number after that would be a 5 space object bound by 6
whachamacallits, etc. etc.

Can anyone prove or disprove this conjecture?

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