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
Crossrefs
Cf. A167028.
Programs
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Mathematica
Rest[Rest[CoefficientList[Series[Sqrt[Cosh[x]]*E^(x^2/4), {x, 0, 20}], x] * Range[0, 20]!]] (* Vaclav Kotesovec, Feb 15 2015 *)
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
Extensions
More terms from Vaclav Kotesovec, Feb 15 2015
Comments