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 A235373 E.g.f. satisfies: A'(x) = A(x)^6 * A(-x)^3 with A(0) = 1. 6

%I

%S 1,1,3,27,249,4041,63243,1475667,32699889,993349521,28523262483,

%T 1066359584907,37641671773929,1670094388871001,69986872318116123,

%U 3592579308449406147,174344892287659801569,10161108739424329621281,560542564223660451017763,36558288488418607271489787

%N E.g.f. satisfies: A'(x) = A(x)^6 * A(-x)^3 with A(0) = 1.

%H Vaclav Kotesovec, <a href="/A235373/b235373.txt">Table of n, a(n) for n = 0..370</a>

%F E.g.f.: 1/(1 - 2*Series_Reversion( Integral (1 - 4*x^2)^(3/2) dx ))^(1/2).

%F Limit n->infinity (a(n)/n!)^(1/n) = 32/(3*Pi) = 3.3953054526271... - _Vaclav Kotesovec_, Jan 29 2014

%F a(n) ~ n! * 2^(3/10) * (32/(3*Pi))^(n+1/5) / (GAMMA(1/5) * 5^(1/5) * n^(4/5)). - _Vaclav Kotesovec_, Jan 30 2014

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 27*x^3/3! + 249*x^4/4! + 4041*x^5/5! +...

%e Related series.

%e A(x)^6 = 1 + 6*x + 48*x^2/2! + 552*x^3/3! + 8064*x^4/4! + 146016*x^5/5! +...

%e Note that 1 - 1/A(x)^2 is an odd function:

%e 1 - 1/A(x)^2 = 2*x + 24*x^3/3! + 2592*x^5/5! + 768384*x^7/7! +...

%e where Series_Reversion((1 - 1/A(x)^2)/2) = Integral (1-4*x^2)^(3/2) dx.

%t CoefficientList[1/(1 - 2*InverseSeries[Series[Integrate[(1 - 4*x^2)^(3/2),x],{x,0,20}],x])^(1/2),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)^3 +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/(1-2*serreverse(intformal((1-4*x^2 +x*O(x^n))^(3/2))))^(1/2); n!*polcoeff(A, n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A235370, A235371, A235372, A235374.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 07 2014

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