OFFSET
0,3
REFERENCES
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.
I. M. H. Etherington, Some problems of non-associative combinations, Edinburgh Math. Notes, 32 (1940), 1-6.
T. Muir, The Theory of Determinants in the Historical Order of Development. 4 vols., Macmillan, NY, 1906-1923, Vol. 3, p. 282.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..225
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. [Annotated scanned copy]
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923, Vol. 2.
T. Muir, The Theory of Determinants in the Historical Order of Development, 4 vols., Macmillan, NY, 1906-1923. [Annotated scans of selected pages] See Vol. 3, page 282.
Wikipedia, Skew-symmetric matrix
FORMULA
a(n) = (2*n)! * [x^(2*n)] (1-x^2)^(-1/4)*exp(x^2/4).
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
a(n) ~ (2*n)! * exp(1/4) * GAMMA(3/4) / (Pi * sqrt(2) * n^(3/4)). - Vaclav Kotesovec, Feb 15 2015
MAPLE
a:= proc(n) option remember;
`if`(n<2, 1, (2*n-1)^2 * a(n-1) -3*binomial(2*n-1, 3) *a(n-2))
end:
seq(a(n), n=0..20);
MATHEMATICA
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 *)
PROG
(PARI)
x='x+O('x^50); v=Vec( (1-x)^(-1/4)*exp(x/4) );
vector(#v, n, v[n]*(2*n-2)! ) \\ Joerg Arndt, Jan 21 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Jon E. Schoenfield, Mar 24 2010
Edited by Alois P. Heinz, Jan 21 2011
STATUS
approved