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@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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