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A027363 Generalizing the 27 lines on a cubic surface: number of lines on the generic hypersurface of degree 2n-1 in complex projective (n+1)-space.

%I A027363 #41 Jan 31 2022 18:57:45
%S A027363 1,27,2875,698005,305093061,210480374951,210776836330775,
%T A027363 289139638632755625,520764738758073845321,1192221463356102320754899,
%U A027363 3381929766320534635615064019,11643962664020516264785825991165
%N A027363 Generalizing the 27 lines on a cubic surface: number of lines on the generic hypersurface of degree 2n-1 in complex projective (n+1)-space.
%D A027363 Van der Waerden, see one of his 'Zur algebraischen Geometrie' papers.
%H A027363 Gheorghe Coserea, <a href="/A027363/b027363.txt">Table of n, a(n) for n = 1..300</a>
%H A027363 Steven R. Finch, <a href="/A013587/a013587.pdf">Enumerative geometry</a>, February 24, 2014. [Cached copy, with permission of the author]
%H A027363 Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 752.
%H A027363 Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, <a href="http://arxiv.org/abs/math.NT/0610286">Sequences of enumerative geometry: congruences and asymptotics</a>, arXiv math.NT/0610286, 2006.
%F A027363 Let b(n, i)=i/(n-i+1) and g(n, k)=s[ k ](b(n, 1), b(n, 2), ..., b(n, n)), where s[ k ] is the k-th elementary symmetric function; a(n) = (2n-1)^2 * (2n-2)! * [ g(2n-2, n-1) - g(2n-2, n) ].
%F A027363 a(n) = [x^n] (1-x)*Product_{j=0..2n-1}(2n-1-j+j*x). [Van der Waerden]
%F A027363 a(n) ~ sqrt(27/Pi) * (2*n-1)^(2*n-3/2) * (1-9/(8*n)+O(1/n^2)). - _Gheorghe Coserea_, Jul 28 2016
%t A027363 a[n_] := Coefficient[ (1-x)*Product[ 2n-1-j+j*x, {j, 0, 2n-1}], x, n]; Table[a[n], {n, 1, 12}] (* _Jean-François Alcover_, Jan 23 2012, from second formula *)
%o A027363 (PARI)
%o A027363 a(n) = my(x='x); polcoeff((1-x) * prod(j=0, 2*n-1, 2*n-1-j + j*x), n);
%o A027363 vector(20, n, a(n))  \\ _Gheorghe Coserea_, Jul 28 2016
%Y A027363 Cf. A013587, A076912.
%K A027363 nonn,nice
%O A027363 1,2
%A A027363 Paolo Dominici (pl.dm(AT)libero.it), Oct 15 1997