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%I #11 Sep 12 2015 11:00:28
%S 1,2,12,108,1320,20400,381360,8366400,210712320,5991572160,
%T 189846961920,6632804344320,253310120743680,10498203901785600,
%U 469251125818675200,22501933753870771200,1152276591132072806400,62756742945031098163200,3622251744055050294988800
%N E.g.f. satisfies: A(x) = x + log(1 + A(x)^2).
%H Vaclav Kotesovec, <a href="/A218652/b218652.txt">Table of n, a(n) for n = 1..375</a>
%H V. Kotesovec, <a href="https://oeis.org/wiki/User:Vaclav_Kotesovec">Asymptotic of implicit functions if Fww = 0</a>, Jan 19 2014
%F E.g.f. A(x) satisfies:
%F (1) A(x - log(1+x^2)) = x.
%F (2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) log(1+x^2)^n / n!.
%F (3) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) log(1+x^2)^n/x / n! ).
%F a(n) = n*A218653(n-1).
%F a(n) ~ GAMMA(1/3) * n^(n-5/6) / (6^(1/6) * sqrt(Pi) * exp(n) * (1-log(2))^(n-1/3)). - _Vaclav Kotesovec_, Jan 19 2014
%e E.g.f: A(x) = x + 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1320*x^5/5! +...
%e Related series:
%e A(x)^2 = 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1560*x^5/5! + 25200*x^6/6! +...
%e log(1+A(x)^2) = 2*x^2/2! + 12*x^3/3! + 108*x^4/4! + 1320*x^5/5! +...
%t Rest[CoefficientList[InverseSeries[Series[x - Log[1 + x^2],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 19 2014 *)
%o (PARI) {a(n)=n!*polcoeff(serreverse(x-log(1+x^2 +x*O(x^n))), n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, log(1 + x^2+x*O(x^n))^m)/m!); n!*polcoeff(A, n)}
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, log(1 + x^2+x*O(x^n))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
%Y Cf. A218653, A213640.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Nov 03 2012
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