[seqfan] Re: Comtets triangles in convex polygons
njasloane at gmail.com
Thu Nov 29 15:30:31 CET 2018
In my copy of Comtet's 1974 book I have a pencilled note in the
margin saying "not integral"
All the other sequences have A-numbers written next to them,
but because these were not integers I did not add them to the OEIS.
I also have the original 2-volume French edition, Analyse combinatoire,
which has the same Problem 8,
but it does not have the sub-problem 8(5).
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 at gmail.com
On Thu, Nov 29, 2018 at 5:14 AM Richard J. Mathar <mathar at mpia-hd.mpg.de>
> Comtet's book of 1974 has a dubious expression on page 75,
> exercise 8(5), as spelled out in
> http://oeis.org/wiki/Sequences_from_Comtet%27s_Advanced_Combinatorics .
> This deals with triangles inside convex polygons constructed by diagonals.
> What is the correct result? Do we just have to wrap the entire
> polynomial in a round() to get correct numbers, which would
> have been written as ||n_3(n^3+18*n^2+43*n+60)/6!|| with two double
> bars in the book, to get 1, 7, 31, 97, 247,... for n>=1? This is not
> correct, because for n<=3 there are no diagonals in the polygons, so there
> none of these triangles.
> That sort of error could be guessed from a similar observation in A321988.
> Or are there other typos, as in A321986?
> I'd expect to be 4 of these triangles in the 4-gon, defined by the
> two ways of defining a diagonal, and each of these has such a triangle
> on either side. This naively produces n*(n-3)/2 = A000096(n-3) for n>4,
> since there are n points for the start of a diagonal, n-3 candidates for
> the other end of the diagonal, a factor of 2 for the two sides of
> the diagonal if n=4, and a division through 2 for the double counting
> of the two directions of the diagonal. This is obviously not what
> Comtet had in mind.
> (Link to the book: at the end of that OEIS wiki page.
> The Pochhammer symbol (n)_3 is introduced on page 6.
> Related sequences A217753, A217754, A277652)
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