A213068
Expansion of e.g.f. arcsinh(sech(x)^2), even powers only.
Original entry on oeis.org
0, -1, 5, -31, -85, 4919, 1248125, -158970631, 2094813635, 2311829506319, -210731879464555, -109642894428121231, 37051431528058442555, 4409666909576599299719, -6492299377660512249146035, 648925901618982079024132169
Offset: 0
(arcsinh(sech(x)^2) - arcsinh(1))/sqrt(2) = -x^2/2 + 5*x^4/4! - 31*x^6/6! - 85*x^8/8! + ...
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Part[#, Range[1, Length[#], 2]] &@(Array[#! &, Length[#], 0]*#) &@ CoefficientList[Series[(ArcSinh[Sech[x]^2] - ArcSinh[1])/Sqrt[2], {x, 0, 30}], x] // ExpandAll
A213067
E.g.f.: arctan(cos(x)^2) - Pi/4.
Original entry on oeis.org
0, -1, -2, 44, 1408, -18016, -5095232, -139605376, 56961507328, 8306292414464, -1178066937638912, -640316054325354496, -7088737339266301952, 76268423227563817631744, 18895160315230467816030208, -12297988177132848140606242816
Offset: 0
arctan(cos(x)^2) - Pi/4 = 0 - x^2/2 - 2*x^4/4! + 44*x^6/6! + 1408*x^8/8! + ...
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Part[#, Range[1, Length[#], 2]] &@(Array[#! &, Length[#], 0]*#) &@
CoefficientList[Series[ArcTan[Cos[x]^2] - Pi/4, {x, 0, 30}], x] // ExpandAll
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 *)
A213069
Expansion of e.g.f. arcsinh(cos(x)*sech(x)), even powers only.
Original entry on oeis.org
0, -1, 3, -1, -77, -13921, 791043, 23892959, -3518362637, -801110007361, 149920222346883, 24069808471917119, -7334638751184472397, -2673575321959933341601, 1059696929013386749787523, 413637485668406346391368479
Offset: 0
(arcsinh(cos(x)*sech(x))-arcsinh(1))/sqrt(2) = -x^2/2 + 3*x^4/4! - x^6/6! - 77*x^8/8! + ...
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Part[#, Range[1, Length[#], 2]] &@(Array[#! &, Length[#], 0]*#) &@ CoefficientList[Series[(ArcSinh[Cos[x]*Sech[x]] - ArcSinh[1])/Sqrt[2], {x, 0, 30}], x] // ExpandAll
With[{nn=30},Take[CoefficientList[Series[(ArcSinh[Cos[x]Sech[x]]-ArcSinh[ 1])/ Sqrt[2],{x,0,nn}],x]Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Mar 24 2013 *)
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