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

1,4

Comments

If n is odd a(n)=0.

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

Links

Table of n, a(n) for n=1..31.

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)).

Example

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).

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 *)

Crossrefs

Cf. A002370, A167028, A002135.

Sequence in context: A187525 A187696 A005212 * A246137 A052679 A266218

Adjacent sequences: A167025 A167026 A167027 * A167029 A167030 A167031

Keyword

nonn,easy

Author

Pietro Majer, Oct 27 2009

Status

approved