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
KEYWORD
nonn,nice
AUTHOR
Paolo Dominici (pl.dm(AT)libero.it), Oct 15 1997
STATUS
approved