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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. 3
1, 27, 2875, 698005, 305093061, 210480374951, 210776836330775, 289139638632755625, 520764738758073845321, 1192221463356102320754899, 3381929766320534635615064019, 11643962664020516264785825991165 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
Van der Waerden, see one of his 'Zur algebraischen Geometrie' papers.
LINKS
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
Sequence in context: A050644 A048567 A227492 * A350135 A085529 A132659
KEYWORD
nonn,nice
AUTHOR
Paolo Dominici (pl.dm(AT)libero.it), Oct 15 1997
STATUS
approved

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Last modified June 21 14:37 EDT 2024. Contains 373547 sequences. (Running on oeis4.)