A167029 Difference between the number of positive and negative terms in the expansion of a skew symmetric matrix of order n.

1, 0, 2, 0, 8, 0, 18, 0, 578, 0, -15460, 0, 1012512, 0, -81237604, 0, 8572174172, 0, -1139408178984, 0, 186543348044576, 0, -36888247922732008, 0, 8669441321229610968, 0, -2388740252077518073072, 0, 762715125987833507921408, 0, -279382350611903941569174000, 0

Offset

1,3

Comments

For even n, a(n)=0.

Links

Table of n, a(n) for n=1..32.

Formula

E.g.f. (for offset 2): sqrt(cosh(x))*exp(x^2/4).

Asymptotics (for even n): a(n)=exp(Pi^2/16)*(2^(n-2))*(n!)*(Pi^(-n))*n^(3/4)*(1+O(1/n)) [This formula is wrong. - Vaclav Kotesovec, Feb 15 2015]

If n is odd |a(n)| ~ exp(-Pi^2/16) * 2^(n+1/2) * n! / (sqrt(n) * Pi^(n+1)). - Vaclav Kotesovec, Feb 15 2015

Mathematica

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

Crossrefs

Cf. A167028.

Sequence in context: A073830 A213272 A053205 * A094030 A199573 A103424

Adjacent sequences: A167026 A167027 A167028 * A167030 A167031 A167032

Keyword

easy,nice,sign

Author

Pietro Majer, Oct 27 2009

Extensions

More terms from Vaclav Kotesovec, Feb 15 2015

Status

approved