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Expansion of e.g.f. tan(arcsinh(x)) = cot(arccosh(x)).
1

%I #13 Jul 22 2018 09:24:02

%S 1,1,5,5,1865,-111575,15280525,-2577345875,591476692625,

%T -172699076899375,62926850998383125,-27954459594642936875,

%U 14878753146153329265625,-9347233537301419814234375

%N Expansion of e.g.f. tan(arcsinh(x)) = cot(arccosh(x)).

%C With changed signs, expansion of e.g.f. tanh(arcsin(x)).

%H Muniru A Asiru, <a href="/A102060/b102060.txt">Table of n, a(n) for n = 1..60</a>

%F a(n) ~ (-1)^(n+1) * 2^(2*n) * n^(2*n-2) / ((cosh(Pi)+1)*exp(2*n)). - _Vaclav Kotesovec_, Oct 23 2013

%e tan(arcsinh(x)) = x + x^3/3! + 5x^5/5! + 5x^7/7! + 1865x^9/9! - ...

%p seq(coeff(series(factorial(n)*tan(arcsinh(x)), x,n+1),x,n),n=1..30,2); # _Muniru A Asiru_, Jul 22 2018

%t Table[n!*SeriesCoefficient[Tan[ArcSinh[x]],{x,0,n}],{n,1,40,2}] (* _Vaclav Kotesovec_, Oct 23 2013 *)

%K sign

%O 1,3

%A _Ralf Stephan_, Dec 28 2004