# [seqfan] Re: The Fibonomial Triangle

Richard Guy rkg at cpsc.ucalgary.ca
Sat Dec 4 23:41:15 CET 2010

```Maximilian,
That's more like it!  Even better, I now
see, is A055870.  I wonder if all the cross-references
between all the sequences mentioned are mentioned?  R.

On Sat, 4 Dec 2010, Maximilian Hasler wrote:

> some more (or less) information might be in:
> http://oeis.org/A010048 :	Triangle of Fibonomial coefficients.
>
> Maximilian
>
> On Sat, Dec 4, 2010 at 5:20 PM, Richard Guy <rkg at cpsc.ucalgary.ca> wrote:
>> Dear all,
>>          How much of the following is known to those
>> who well know it?  I haven't yet been able to consult
>> Knuth, vol.1, p.85, so it may be there.  Or in Duke
>> Math J 29(1962) page numbers need correcting in some?
>> of [A000012, A000045, A007598], A056570--4, A056585--7.
>>
>> The characteristic polynomials for these sequences are
>>
>>                             x - 1
>>                          x^2 - x - 1
>>                      x^3 - 2x^2 - 2x + 1
>>                  x^4 - 3x^3 - 6x^2 + 3x + 1
>>              x^5 - 5x^4 - 15x^3 + 15x^2 + 5x - 1
>>          x^6 - 8x^5 - 40x^4 + 60x^3 + 40x^2 - 8x - 1
>>   x^7 - 13x^6 - 104x^5 + 260x^4 + 260x^3 - 104x^2 - 13x + 1
>>           x^8 -21 -273 +1092 +1820 -1092 -273 +21 +1
>> x^9 -55 -1870 +19635 +85085 -136136 -85086 +19635 +1870 -55 -1
>> x^10 -89 -4895 +83215 +582505 -1514513 -1514513 +582505 + - - +
>>
>> which (it is known by some) factor into
>>                            x-1
>>                          x^2-x-1
>>                      (x+1)(x^2-3x+1)
>>                    (x^2+x-1)(x^2-4x-1)
>>                 (x-1)(x^2+3x+1)(x^2-7x+1)
>>              (x^2-x-1)(x^2+4x-1)(x^2-11x-1)
>>           (x+1)(x^2-3x+1)(x^2+7x+1)(x^2-18x+1)
>>         (x^2+x-1)(x^2-4x-1)(x^2+11x-1)(x^2-29x-1)
>>      (x-1)(x^2+3x+1)(x^2-7x+1)(x^2+18x+1)(x^2-47x+1)
>>    (x^2-x-1)(x^2+4x-1)(x^2-11x-1)(x^2+29x-1)(x^2-76x-1)
>> (x+1)(x^2-3x+1)(x^2+7x+1)(x^2-18x+1)(x^2+47x+1)(x^2-123x+1)
>>
>> where the middle coefficients are, of course, Lucas numbers.
>>
>> If we make a triangle of the coeffs of the unfactored
>> polynomials, we find that, apart from signs, they are
>>
>>                                 e             (F0/F1)
>>
>>                           e             F1/F1
>>
>>                      e        F2/F1          F2F1/F1F2
>>
>>                  e       F3/F1         F3F2/F1F2      F3F2F1/F1F2F3
>>
>>             e     F4/F1     F4F3/F1F2      F4F3F2/F1F2F3  F4F3F2F1/F1F2F3F4
>>
>>         e     F5/F1    F5F4/F1F2     F5F4F3/F1F2F3  ....
>>
>>     e    F6/F1  F6F5/F1F2 F6F5F4/F1F2F3  F6F5F4F3/F1F2F3F4  ''''
>>
>> e    F7/F1 F7F6/F1F2 F7F6F5/F1F2F3 F7F6F5F4/F1F2F3F4  ....
>>
>> where  e = 1  is the empty product, and F7F6F5/F1F2F3,
>> for example, is 13*8*5/1*1*2 = 260, a sort of `binomial coefficient'
>> of Fibonacci numbers F1=F2=1, F3=2, F4=3, ...
>>
>> Of course, these sequences are all divisibility sequences.    R.
>>
>>
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```