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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