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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 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

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