Finding the Pattern of Triangular Set of Numbers

[isthereaspoon]

I have encountered such a triangular set of numbers:

1
-8 -7
28 -16 9
-56 -65 -22 -7
70 112 51 4 1

Apart from the obvious first column being (x-1)^8, what can you find for other columns?

 

[CRGreathouse]

I can't see anything. Sloane's suggests
http://oeis.org/classic/A130595
which is tantalizing but I can't find the other parts.

 

[isthereaspoon]

thanks, it is simply hard.

 

[squelchy451]

The 'hypothenuse' seems to play a role in that

1
-8 -7
28 -16 9

-7 - 1 = -8
-7 - 9 = -16

As far as the other numbers go I haven't gotten them yet, but just putting this out there as a possible stepping stone

 

[blue_raver22]

I find this triangular set of numbers interesting and would like to investigate further. First, I would like to clarify that the first column is not actually (x-1)^8, but rather the sequence of triangular numbers (1, 3, 6, 10, 15, ...).

Moving on to the other columns, I can see that the second column is formed by alternating positive and negative multiples of 8, starting with -8 and increasing by 8 for each subsequent number. This pattern could be expressed as (-1)^n * 8 * n, where n is the row number.

Similarly, the third column is formed by alternating positive and negative multiples of 7, starting with -7 and increasing by 7 for each subsequent number. This pattern could be expressed as (-1)^n * 7 * (n-1).

The fourth column follows a more complex pattern, with the numbers increasing in magnitude but alternating between positive and negative. After some analysis, I have found that the pattern can be expressed as (-1)^n * (n^2 - 4n + 5).

Finally, the fifth column seems to be a combination of the previous patterns, with the numbers increasing in magnitude and alternating between positive and negative. This can be expressed as (-1)^n * (n^3 - 6n^2 + 11n - 6).

In summary, the pattern for the triangular set of numbers can be described as:

Column 1: Triangular numbers (1, 3, 6, 10, 15, ...)
Column 2: (-1)^n * 8 * n
Column 3: (-1)^n * 7 * (n-1)
Column 4: (-1)^n * (n^2 - 4n + 5)
Column 5: (-1)^n * (n^3 - 6n^2 + 11n - 6)

Further exploration and analysis may uncover more patterns and relationships within this set of numbers. This could potentially lead to a deeper understanding of triangular numbers and their properties. Thank you for bringing this interesting set of numbers to my attention.