[seqfan] Comtets triangles in convex polygons
Richard J. Mathar
mathar at mpia-hd.mpg.de
Thu Nov 29 11:13:57 CET 2018
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 are
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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