A012011 Expansion of e.g.f. arcsinh(sin(sin(x))), odd powers only.

1, -3, 41, -1515, 113489, -14321299, 2716798969, -721988679547, 255890614389921, -116619928299347491, 66439023428917759945, -46262911349611152912395, 38656874720522348065089521

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Formula

a(n) ~ (-1)^n * c * d^(2*n) * (2*n)! / sqrt(n), where d = 1 / arcsinh(arcsinh(1)) = 1.25792845770279367616342387910366384..., c = 0.8688030872074338111899369246534915... . - Vaclav Kotesovec, Feb 03 2015

Example

x-3/3!*x^3+41/5!*x^5-1515/7!*x^7+113489/9!*x^9...

Mathematica

Take[Rest[With[{nn=25}, CoefficientList[Series[ArcSinh[Sin[Sin[x]]], {x, 0, nn}], x]Range[0, nn]!]], {1, -1, 2}] (* Harvey P. Dale, Oct 21 2011 *)
(* Alternative: *)
nmax = 15; CoefficientList[Series[(Cos[Sqrt[x]]*Cos[Sin[Sqrt[x]]]) / Sqrt[1 + Sin[Sin[Sqrt[x]]]^2], {x, 0, nmax}], x] * (2*Range[0, nmax])! (* Vaclav Kotesovec, Aug 25 2025 *)

Crossrefs

Sequence in context: A187540 A012104 A012147 * A277453 A147797 A139161

Adjacent sequences: A012008 A012009 A012010 * A012012 A012013 A012014

Keyword

sign,changed

Author

Patrick Demichel

Extensions

Confirmed by Harvey P. Dale, Oct 21 2011

Name edited by Michel Marcus, Aug 25 2025

Status

approved