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%I #11 Nov 21 2017 18:44:49
%S 1,1,5,53,745,14761,342125,9919133,320525905,12231596881,507077877845,
%T 23997083985413,1214302628958265,68589494705086201,
%U 4098200887304658365,269019686917890755693,18535055580911326897825,1386438585493229417193121,108209001114760469546923685
%N E.g.f. satisfies: A'(x) = A(x)^6 * A(-x) with A(0) = 1.
%H G. C. Greubel, <a href="/A235371/b235371.txt">Table of n, a(n) for n = 0..358</a>
%F E.g.f.: 1/(1 - 4*Series_Reversion( Integral (1 - 16*x^2)^(1/4) dx ))^(1/4).
%F Lim_{n->infinity} (a(n)/n!)^(1/n) = 12*sqrt(2) * GAMMA(3/4)^2 / Pi^(3/2) = 4.57655858101... - _Vaclav Kotesovec_, Jan 28 2014
%e E.g.f.: A(x) = 1 + x + 5*x^2/2! + 53*x^3/3! + 745*x^4/4! + 14761*x^5/5! + ...
%e Related series.
%e A(x)^6 = 1 + 6*x + 60*x^2/2! + 888*x^3/3! + 17040*x^4/4! + 407136*x^5/5! + ...
%e Note that 1 - 1/A(x)^4 is an odd function:
%e 1 - 1/A(x)^4 = 4*x + 32*x^3/3! + 4864*x^5/5! + 2250752*x^7/7! + ...
%e where Series_Reversion((1 - 1/A(x)^4)/4) = Integral (1-16*x^2)^(1/4) dx.
%t CoefficientList[1/(1 - 4*InverseSeries[Series[Integrate[(1-16*x^2)^(1/4), x],{x,0,20}],x])^(1/4),x] * Range[0,20]! (* _Vaclav Kotesovec_, Jan 28 2014 *)
%o (PARI) {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^6*subst(A, x, -x)^1 +x*O(x^n) )); n!*polcoeff(A, n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=1); A=1/(1-4*serreverse(intformal((1-16*x^2 +x*O(x^n))^(1/4))))^(1/4); n!*polcoeff(A, n)}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A235370, A235372, A235373, A235374.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 07 2014
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