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%I #16 Jan 13 2014 10:44:49
%S 1,1,4,25,231,2701,38963,662929,13037608,290586301,7241964357,
%T 199511192233,6020966627261,197526938563501,6999280556977816,
%U 266408120037084577,10840080711977589375,469561023814159909981,21573682777922810043335,1047866254345761285979321
%N E.g.f. is series reversion of log(1+x)*sqrt(1-x^2).
%C Compare to: Series_Reversion(log(1+x)*(1+x)) = Sum_{n>=1} -(n-1)^(n-1)*(-x)^n/n!.
%H Paul D. Hanna, <a href="/A224080/b224080.txt">Table of n, a(n) for n = 1..200</a>
%F E.g.f. A(x) satisfies: 1 + A(x) = exp( x / sqrt(1 - A(x)^2) ).
%F a(n) ~ n^(n-1) * (s/((1-s)*sqrt(1-s^2)))^n * ((1-s^2)/sqrt(1+s+s^2)) / exp(n), where s = 0.66288612806066020129... is the root of the equation s*log(1+s) = 1-s. - _Vaclav Kotesovec_, Jan 13 2014
%e E.g.f.: A(x) = x + x^2/2! + 4*x^3/3! + 25*x^4/4! + 231*x^5/5! + 2701*x^6/6! +...
%t Rest[CoefficientList[InverseSeries[Series[Log[1+x]*Sqrt[1-x^2], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 13 2014 *)
%o (PARI) {a(n)=local(X=x+x*O(x^n));n!*polcoeff(serreverse(log(1+X)*sqrt(1-X^2)), n)}
%o for(n=1,25,print1(a(n),", "))
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jul 20 2013
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