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%I #16 Sep 08 2022 08:45:16
%S 1,4,80,3680,300800,38233600,6960179200,1715803648000,549927772160000,
%T 222172054667264000,110418310176112640000,66193620737183580160000,
%U 47093975714823012352000000,39225810499489538768896000000,37809201187185822383734784000000,41754105224509634604034949120000000
%N Expansion of e.g.f. tan(arctanh(x)), odd powers only.
%C With alternating signs, expansion of tanh(arctan(x)).
%H Vincenzo Librandi, <a href="/A102063/b102063.txt">Table of n, a(n) for n = 1..100</a>
%e tan(arctanh(x)) = x + 4*x^3/3! + 80*x^5/5! + 3680*x^7/7! + 300800*x^9/9! + ...
%p seq(coeff(series(factorial(n)*tan(arctanh(x)), x, n+1), x, n), n = 1 .. 32, 2); # _Muniru A Asiru_, Aug 16 2018
%t nmax=20; Table[(CoefficientList[Series[Tan[ArcTanh[x]], {x, 0, 2*nmax}], x] Range[0, 2 nmax - 1]!)[[n]], {n, 2, 2 nmax, 2}] (* _Vincenzo Librandi_, Aug 16 2018 *)
%o (Magma) m:=35; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1+Tan(Argtanh(x)))); [Factorial(n-1)*b[n]: n in [2..m by 2]]; // _Vincenzo Librandi_, Aug 16 2018
%K nonn
%O 1,2
%A _Ralf Stephan_, Dec 28 2004
%E Example corrected by _Vaclav Kotesovec_, Aug 16 2018
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