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.

1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, 289139638632755625, 520764738758073845321, 1192221463356102320754899, 3381929766320534635615064019, 11643962664020516264785825991165

Offset

1,2

References

Van der Waerden, see one of his 'Zur algebraischen Geometrie' papers.

Links

Gheorghe Coserea, Table of n, a(n) for n = 1..300

Steven R. Finch, Enumerative geometry, February 24, 2014. [Cached copy, with permission of the author]

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 752.

Daniel B. Grunberg and Pieter Moree, with an Appendix by Don Zagier, Sequences of enumerative geometry: congruences and asymptotics, arXiv math.NT/0610286, 2006.

Formula

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) ].

a(n) = [x^n] (1-x)*Product_{j=0..2n-1}(2n-1-j+j*x). [Van der Waerden]

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

Mathematica

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 *)

Prog

(PARI)
a(n) = my(x='x); polcoeff((1-x) * prod(j=0, 2*n-1, 2*n-1-j + j*x), n);
vector(20, n, a(n))  \\ Gheorghe Coserea, Jul 28 2016

Crossrefs

Cf. A013587, A076912.

Sequence in context: A050644 A048567 A227492 * A350135 A085529 A132659

Adjacent sequences: A027360 A027361 A027362 * A027364 A027365 A027366

Keyword

nonn,nice

Author

Paolo Dominici (pl.dm(AT)libero.it), Oct 15 1997

Status

approved