A027363, 27 lines on a cubic surface etc.

[Pieter Moree]

Dear sequence fan,

Sequence A027363 generalises the 27 lines on a cubic surface:
the nth term is defined as the number of lines on the generic hypersurface
of degree 2n-1 in complex projective (n+1)-space.
It starts like:
1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775,
289139638632755625, 520764738758073845321, 1192221463356102320754899,
3381929766320534635615064019,...

It was proved by Van der Waerden in one of his famous
`Zur algebraischen Geometrie' papers (he wrote about 20 in total
with this title I believe) that

v_n is the coefficient of x^{n-1} in the polynomial

(1-X)\prod_{j=0}^{2n-3}(2n-3-j+jX)

Daniel Grunberg discovered many congruence properties of the sequence v_n.
[I am pretty sure there is much more to be discovered in this regard.]

Don Zagier proved an intriguing asymptotic formula for the v_n and
gave a completely different proof of Van der Waerden's identity.

I showed that Dominici's formula given at OEIS for
v_n can be derived from Van der Waerden's.

Soon a preprint on this:
Daniel B. Grünberg, Pieter Moree. Appendix by Don Zagier.

will be brought out on the arXiv.

For a pre-preprint see
http://guests.mpim-bonn.mpg.de/moree/inprep.html

** Pieter Moree **