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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]

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-Fran├žois 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 * A085529 A132659 A104131

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 April 9 22:10 EDT 2020. Contains 333371 sequences. (Running on oeis4.)