A167028 Number of terms in the expansion of the determinant of a skew-symmetric matrix of order n.

0, 1, 0, 6, 0, 120, 0, 5250, 0, 395010, 0, 45197460, 0, 7299452160, 0, 1580682203100, 0, 441926274289500, 0, 154940341854097800, 0, 66565404923242024800, 0, 34389901168124209507800, 0, 21034386936107260971255000, 0, 15032296693671903309613950000, 0, 12411582569784462888618434640000, 0

Offset: 1

Author

Pietro Majer, Oct 27 2009

Keywords

• nonn
• easy

Comments

If n is odd a(n)=0.

Essentially a duplicate of A002370. - N. J. A. Sloane, Oct 27 2009

Examples

			Example: the determinant of a skew symmetric matrix of order n=4 is
det(A)=A(1,2)A(1,2)A(3,4)A(3,4) + 2A(1,2)A(2,3)A(1,4)A(3,4) -2A(1,2)A(2,4)A(1,3)A(3,4)+ A(1,3)A(1,3)A(2,4)A(2,4)-2A(1,3)A(2,4)A(1,4)A(2,3)+A(1,4)A(1,4)A(2,3)A(2,3).
		

Crossrefs

Cf. A002370, A167028, A002135.

Programs

Maple

for n from 1 to 20 do a[n]:=n!coeftayl( (1-x^2)^(-1/4)*exp(x^2/4),x=0,n) od;

Mathematica

Rest[CoefficientList[Series[(1-x^2)^(-1/4)*E^(x^2/4), {x, 0, 20}], x] * Range[0, 20]!] (* Vaclav Kotesovec, Feb 15 2015 *)

Formula

Exponential generating function: (1-x^2)^(-1/4) exp(x^2/4).

Asymptotics (for even n): a(n)= (n!/Pi)exp( (-3log(n)+1+log(2))/4 ) GAMMA(3/4) (1+O(1/n)). [corrected by Vaclav Kotesovec, Feb 15 2015]. More elegant form is a(n) ~ n! * 2^(1/4) * exp(1/4) * GAMMA(3/4) / (Pi * n^(3/4)).