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%I M4296 N1796 #50 Dec 28 2022 09:04:25
%S 1,1,6,120,5250,395010,45197460,7299452160,1580682203100,
%T 441926274289500,154940341854097800,66565404923242024800,
%U 34389901168124209507800,21034386936107260971255000,15032296693671903309613950000,12411582569784462888618434640000
%N a(n) = (2*n-1)^2 * a(n-1) - 3*C(2*n-1,3) * a(n-2) for n>1; a(0) = a(1) = 1.
%D A. C. Aitken, On the number of distinct terms in the expansion of symmetric and skew determinants, Edinburgh Math. Notes, No. 34 (1944), 1-5.
%D I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.
%D T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A002370/b002370.txt">Table of n, a(n) for n = 0..225</a>
%H A. C. Aitken, <a href="/A002135/a002135_2.pdf">On the number of distinct terms in the expansion of symmetric and skew determinants</a>, Edinburgh Math. Notes, No. 34 (1944), 1-5. [Annotated scanned copy]
%H T. Muir, <a href="http://www.hti.umich.edu/cgi/t/text/text-idx?sid=b88432273f115fb346725f1a42422e19;c=umhistmath;idno=ACM9350.0003.001">The Theory of Determinants in the Historical Order of Development</a>, 4 vols., Macmillan, NY, 1906-1923, Vol. 2.
%H T. Muir, <a href="/A002135/a002135_1.pdf">The Theory of Determinants in the Historical Order of Development</a>, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages] See Vol. 3, page 282.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Skew-symmetric_matrix">Skew-symmetric matrix</a>
%F a(n) = (2*n)! * [x^(2*n)] (1-x^2)^(-1/4)*exp(x^2/4).
%F a(n) = 2^n*GAMMA(n+1/2)*A002801(n)/Pi^(1/2) = GAMMA(n+1/2)*hypergeom([1/4, -n],[],-4)/Pi^(1/2). - _Mark van Hoeij_, Oct 26 2011
%F a(n) ~ (2*n)! * exp(1/4) * GAMMA(3/4) / (Pi * sqrt(2) * n^(3/4)). - _Vaclav Kotesovec_, Feb 15 2015
%p a:= proc(n) option remember;
%p `if`(n<2, 1, (2*n-1)^2 * a(n-1) -3*binomial(2*n-1, 3) *a(n-2))
%p end:
%p seq(a(n), n=0..20);
%t a[n_] := Gamma[n+1/2]*HypergeometricPFQ[{1/4, -n}, {}, -4]/Sqrt[Pi]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Mar 17 2014, after _Mark van Hoeij_ *)
%o (PARI)
%o x='x+O('x^50); v=Vec( (1-x)^(-1/4)*exp(x/4) );
%o vector(#v,n, v[n]*(2*n-2)! ) \\ _Joerg Arndt_, Jan 21 2011
%Y Cf. A167028.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Jon E. Schoenfield_, Mar 24 2010
%E Edited by _Alois P. Heinz_, Jan 21 2011
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