A213067 - OEIS

%I #15 Feb 24 2022 16:52:38

%S 0,-1,-2,44,1408,-18016,-5095232,-139605376,56961507328,8306292414464,

%T -1178066937638912,-640316054325354496,-7088737339266301952,

%U 76268423227563817631744,18895160315230467816030208,-12297988177132848140606242816

%N E.g.f.: arctan(cos(x)^2) - Pi/4.

%C This function is even, with constant term Pi/4 = 0.785398163397...

%C It was missing from OEIS entries by Patrick Demichel.

%H Vincenzo Librandi, <a href="/A213067/b213067.txt">Table of n, a(n) for n = 0..100</a>

%F E.g.f.: arctan(cos(x)^2) - Pi/4.

%e arctan(cos(x)^2) - Pi/4 = 0 - x^2/2 - 2*x^4/4! + 44*x^6/6! + 1408*x^8/8! + ...

%t Part[#, Range[1, Length[#], 2]] &@(Array[#! &, Length[#], 0]*#) &@

%t   CoefficientList[Series[ArcTan[Cos[x]^2] - Pi/4, {x, 0, 30}], x] // ExpandAll

%t With[{nn=30},Take[CoefficientList[Series[ArcTan[Cos[x]^2]-Pi/4,{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Feb 24 2022 *)

%Y Cf. A213066, A213068, A213069.

%K sign

%O 0,3

%A _Olivier Gérard_, Jun 04 2012