A012025 E.g.f. arcsinh(sin(arctan(x))) = arcsinh(x/(1+x^2)^(1/2)) (odd powers only).

1, -4, 84, -4320, 418320, -66225600, 15657364800, -5187108326400, 2296766568096000, -1310785979158656000, 937056917610253440000, -820081468493365478400000, 862301491174096979765760000

Offset

1,2

Links

Table of n, a(n) for n=1..13.

Formula

E.g.f. arcsinh(sin(arctan(x))) = arcsinh(x/(1+x^2)^(1/2)).

arcsinh(x/(1+x^2)^(1/2)) = x/sqrt(1+x^2)*(1 - x^2/(G(0)+x^2)) where G(k) = 4*k^2 + k*(10+6*x^2) + 5*x^2 + 6 + 2*x^2*(1+x^2)*(k+1)*(2*k+3)^3/G(k+1) ; (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 08 2012

a(n) ~ (-1)^(n+1) * 2^(3*n-1) * n^(2*n-2) / exp(2*n). - Vaclav Kotesovec, Oct 30 2013

Example

arcsinh(sin(arctan(x)))=x-4/3!*x^3+84/5!*x^5-4320/7!*x^7+418320/9!*x^9...

Mathematica

Table[n!*SeriesCoefficient[ArcSinh[x/(1+x^2)^(1/2)], {x, 0, n}], {n, 1, 40, 2}] (* Vaclav Kotesovec, Oct 30 2013 *)

Crossrefs

Sequence in context: A012065 A012139 A012037 * A012107 A012189 A012076

Adjacent sequences: A012022 A012023 A012024 * A012026 A012027 A012028

Keyword

sign

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Extensions

Confirmed by N. J. A. Sloane, Dec 17 2011

Status

approved