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%I #13 Sep 18 2015 17:52:06
%S 0,1,2,6,40,440,5952,97104,1888640,42480000,1082119680,30814080000,
%T 970187827200,33461288899584,1254539018571776,50803163905751040,
%U 2209862882578300928,102761280728930287616,5087062588875762696192,267101579060996713611264
%N E.g.f. satisfies: A(x) = x*(tan(tan(A(x)))+1).
%H Alois P. Heinz, <a href="/A133939/b133939.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) ~ n^(n-1) * (cos(s))^(3/2) * sqrt(s/(-2*r + 2*s + r*sin(2*s))) / (exp(n) * (2*r^2-2*r*s+s^2)^(1/4) * r^(n-3/4)), where r = 0.3331897707672964894... and s = 0.6497639144980735763... are roots of the system of equations r*(sec(s)*sec(tan(s)))^2 = 1, s = r + r*tan(tan(s)). - _Vaclav Kotesovec_, Jul 16 2014
%p A:= proc(n) option remember; if n=0 then 0 else convert(series(x*
%p (tan(tan(A(n-1)))+1), x=0, n+1), polynom) fi
%p end:
%p a:= n-> coeff(A(n), x, n)*n!:
%p seq(a(n), n=0..23);
%t CoefficientList[InverseSeries[Series[x/(1 + Tan[Tan[x]]),{x,0,20}],x],x] * Range[0,20]! (* _Vaclav Kotesovec_, Jul 16 2014 *)
%K nonn
%O 0,3
%A _Alois P. Heinz_, Aug 27 2008
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