A213069 -id:A213069 - cp's OEIS frontend

A213068 Expansion of e.g.f. arcsinh(sech(x)^2), even powers only.

0, -1, 5, -31, -85, 4919, 1248125, -158970631, 2094813635, 2311829506319, -210731879464555, -109642894428121231, 37051431528058442555, 4409666909576599299719, -6492299377660512249146035, 648925901618982079024132169

Offset: 0

Olivier Gérard, Jun 04 2012

sign

This function is even, with constant term arcsinh(1) = 0.881373587019543...

			(arcsinh(sech(x)^2) - arcsinh(1))/sqrt(2) = -x^2/2 + 5*x^4/4! - 31*x^6/6! - 85*x^8/8! + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A213066, A213067, A213069.

Part[#, Range[1, Length[#], 2]] &@(Array[#! &, Length[#], 0]*#) &@ CoefficientList[Series[(ArcSinh[Sech[x]^2] - ArcSinh[1])/Sqrt[2], {x, 0, 30}], x] // ExpandAll

E.g.f.: (arcsinh(sech(x)^2) - arcsinh(1))/sqrt(2).

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

Olivier Gérard, Jun 04 2012

sign

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

It was missing from OEIS entries by Patrick Demichel.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A213066, A213068, A213069.

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 *)

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

cp's OEIS Frontend

A213068 Expansion of e.g.f. arcsinh(sech(x)^2), even powers only.

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