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. -- g