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. (Formerly M4296 N1796)

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

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

Cf. A167028.

Sequence in context: A335334 A094278 A093910 * A012846 A331640 A012641

Adjacent sequences: A002367 A002368 A002369 * A002371 A002372 A002373

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Jon E. Schoenfield, Mar 24 2010

Edited by Alois P. Heinz, Jan 21 2011

Status

approved