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A234292 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x) dx.

%I A234292 #10 Dec 26 2013 09:08:33
%S A234292 1,1,7,103,2365,74305,2970415,144324775,8259853525,544284565825,
%T A234292 40589062689175,3379489549611175,310764664533047725,
%U A234292 31280478682100673025,3420865044191897677375,403897481132212723030375,51205807000808636432381125,6937963969329514181526090625
%N A234292 E.g.f. satisfies: A(x) = 1 + A(x)^4 * Integral 1/A(x) dx.
%F A234292 E.g.f.: 1 + Series_Reversion( x*(6 - 3*x - x^2) / (6*(1+x)^3) ).
%F A234292 E.g.f.: 1 / ( d/dx Series_Reversion( Integral G(x) dx ) ), where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
%F A234292 From _Vaclav Kotesovec_, Dec 26 2013: (Start)
%F A234292 E.g.f.: (sqrt(3)*cos(arccos(5/27 + 64*x/9)/6) - 3*sin(arcsin(sqrt(11/3 - 32*x)/3)/3)) / sqrt(1+6*x)
%F A234292 Recurrence: 11*a(n) = 3*(10*n-37)*a(n-1) + 64*(3*n-5)*(3*n-4)*a(n-2)
%F A234292 a(n) ~ 2^(5*n+3/2) * 3^(n-3) * n^(n+1/2) / (11^(n-1/2) * exp(n) * n^(3/2))
%F A234292 (End)
%e A234292 E.g.f.: A(x) = 1 + x + 7*x^2/2! + 103*x^3/3! + 2365*x^4/4! + 74305*x^5/5! +...
%e A234292 where A( x*(6-3*x-x^2)/(6*(1+x)^3) ) = 1+x.
%e A234292 Related series:
%e A234292 A(x)^4 = 1 + 4*x + 40*x^2/2! + 688*x^3/3! + 17200*x^4/4! + 569680*x^5/5! +...
%e A234292 1/A(x) = 1 - x - 5*x^2/2! - 67*x^3/3! - 1475*x^4/4! - 45265*x^5/5! +...
%t A234292 CoefficientList[1 + InverseSeries[Series[x*(6-3*x-x^2) / (6*(1+x)^3), {x, 0, 20}], x],x]* Range[0, 20]! (* _Vaclav Kotesovec_, Dec 26 2013 *)
%t A234292 FullSimplify[CoefficientList[Series[(Sqrt[3]*Cos[(1/6)*ArcCos[5/27 + (64*x)/9]] - 3*Sin[(1/3)*ArcSin[(1/3)*Sqrt[11/3 - 32*x]]]) / Sqrt[1+6*x], {x, 0, 10}], x] * Range[0, 10]!] (* _Vaclav Kotesovec_, Dec 26 2013 *)
%o A234292 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+A^4*intformal(1/(A^1+x*O(x^n)))); n!*polcoeff(A, n)}
%o A234292 for(n=0, 25, print1(a(n), ", "))
%o A234292 (PARI) {a(n)=local(A=1); A=1+serreverse(x*(6 - 3*x - x^2)/(6*(1+x+x*O(x^n))^3)); n!*polcoeff(A, n)}
%o A234292 for(n=0, 25, print1(a(n), ", "))
%Y A234292 Cf. A234293, A234294.
%K A234292 nonn
%O A234292 0,3
%A A234292 _Paul D. Hanna_, Dec 26 2013