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%I #16 Feb 16 2015 04:10:45
%S 1,0,2,0,8,0,18,0,578,0,-15460,0,1012512,0,-81237604,0,8572174172,0,
%T -1139408178984,0,186543348044576,0,-36888247922732008,0,
%U 8669441321229610968,0,-2388740252077518073072,0,762715125987833507921408,0,-279382350611903941569174000,0
%N Difference between the number of positive and negative terms in the expansion of a skew symmetric matrix of order n.
%C For even n, a(n)=0.
%F E.g.f. (for offset 2): sqrt(cosh(x))*exp(x^2/4).
%F 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]
%F 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
%t Rest[Rest[CoefficientList[Series[Sqrt[Cosh[x]]*E^(x^2/4), {x, 0, 20}], x] * Range[0, 20]!]] (* _Vaclav Kotesovec_, Feb 15 2015 *)
%Y Cf. A167028.
%K easy,nice,sign
%O 1,3
%A _Pietro Majer_, Oct 27 2009
%E More terms from _Vaclav Kotesovec_, Feb 15 2015