# [seqfan] The Fibonomial Triangle

Richard Guy rkg at cpsc.ucalgary.ca
Sat Dec 4 22:20:55 CET 2010

```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.

```