[seqfan] Re: [math-fun] The Fibonomial Triangl

[Richard Guy]

Gareth,
        Thanks for this!  Lucas was much concerned with
the theory of recurring sequences, and is responsible
for the theory, modulo getting it cleaned up by Lehmer,
so it's likely he knew most of what I wrote.  When you
write `no mention there', are you referring to Amer J
Math 1878 or to Knuth?     R.

On Sat, 4 Dec 2010, Gareth McCaughan wrote:

> On Saturday 04 December 2010 21:20:55 Richard Guy 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.
> [etc.]
>
> Knuth vol 1 section 1.2.8 ex  29 reads as follows, aside
> from notation.
>
> 29. [M23] (Fibonomial coefficients.) Edouard Lucas defined
> the quantities (n choose k) sub F = FnF{n-1}...F{n-k+1} / FkF{k-1}...F1
> = prod{j=1..k} F{n-k+j}/F{j} in a manner analogous to
> binomial coefficients. (a) Make a table of (n choose k) sub F
> for 0 <= k <= n <= 6. (b) Show that (n choose k) sub F is always
> an integer because we have (nCk)F = F{k-1} (n-1Ck)F + F{n-k+1}(n-1Ck-1)F.
>
> The solution in Knuth gives the table without comment and
> for (b) simply refers the reader to "E. Lucas, Amer. J. Math.
> 1 (1878), 201--204".
>
> There is no mention there of polynomials, factorization,
> Lucas numbers, or divisibility sequences.