Why stop at two? There are a denumerable number of singularities of Gamma(n+1) / Gamma(k+1) Gamma(n-k+1) at negative integer n , and once you decide to start tinkering with approaching them from multiple randomly selected directions, a whole continuum of choices opens before you --- though admittedly not all of them yield integer results. [ And bear in mind, incidentally, that you will have royally screwed up your symbolic theorem prover --- see examples from Mathematica and Maple in my note. ] WFL On 7/23/17, James Propp <jamespropp@gmail.com> wrote:
Maybe n-choose-k should actually be two-valued (like sqrt(z)), with a branch-point at n=k=0?
Jim Propp
On Saturday, July 22, 2017, Fred Lunnon <fred.lunnon@gmail.com> wrote:
@#$%^&?! ... DropBox is getting steadily less user-friendly --- why give me a link and immediately disable the infernal thing? Anyway, I have copied the file, deleted it and re-up-loaded it: try instead
https://www.dropbox.com/s/9vvl6l6ym1zkret/binomial.pdf
WFL
On 7/23/17, Neil Sloane <njasloane@gmail.com <javascript:;>> wrote:
Fred, when I tried to look at your discussion, https://www.dropbox.com/s/anykne0pd55ehjg/binomial.pdf I got a rude "go away" message
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com <javascript:;>
On Sat, Jul 22, 2017 at 7:20 PM, Fred Lunnon <fred.lunnon@gmail.com <javascript:;>> wrote:
See the note I wrote up following a discussion on this list circa 2014 --- https://www.dropbox.com/s/anykne0pd55ehjg/binomial.pdf
[ I did contemplate trying to get this published in the Math. Intelligencer, but retreated in the face of an editor's mysterious antipathy to computer- generated illustrations. ]
WFL
On 7/22/17, Gareth McCaughan <gareth.mccaughan@pobox.com <javascript:;>> wrote:
I'm guessing that a similar lapse of conviction ruined Mathematica's Binomial function: Some moron complained about the asymmetry so long after it was implemented that WRI forgot why it's asymmetrical, and accommodated the moron.
For those of us who aren't familiar with all Mathematica's quirks, would you care to say what erroneous symmetry Mma gives to the binomial coefficients?
(My apologies if this was already being discussed in some other thread and I just missed it.)
-- g
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