A002370 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.

1, 1, 6, 120, 5250, 395010, 45197460, 7299452160, 1580682203100, 441926274289500, 154940341854097800, 66565404923242024800, 34389901168124209507800, 21034386936107260971255000, 15032296693671903309613950000, 12411582569784462888618434640000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

Crossrefs

Cf. A167028.

Programs

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 *)

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

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

Extensions

More terms from Jon E. Schoenfield, Mar 24 2010

Edited by Alois P. Heinz, Jan 21 2011