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A120593 G.f. satisfies: 5*A(x) = 4 + x + A(x)^4, starting with [1,1,6].

%I A120593 #9 Nov 28 2017 03:36:32
%S A120593 1,1,6,76,1201,21252,402892,8001412,164321982,3461110532,74358814838,
%T A120593 1623152780808,35897318940028,802620009567628,18112759482614328,
%U A120593 412020809942451504,9437537418826749369,217486633306640519124
%N A120593 G.f. satisfies: 5*A(x) = 4 + x + A(x)^4, starting with [1,1,6].
%C A120593 See comments in A120588 for conditions needed for an integer sequence to satisfy a functional equation of the form: r*A(x) = c + b*x + A(x)^n.
%F A120593 G.f. satisfies:
%F A120593 (1) A(x) = 1 + Series_Reversion(1+5*x - (1+x)^4).
%F A120593 (2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (4+x)^(3*n+1)/5^(4*n+1), by Lagrange Inversion.
%F A120593 (3) A(x) = F(x/A(x)) and F(x) = A(x*F(x)) where F(x) = (4 + F(x)^4)/(5-x)  is the g.f. of A244856. - _Paul D. Hanna_, Jul 09 2014
%F A120593 a(n) ~ 2^(-7/3 + 3*n) * (-32 + 15*10^(1/3))^(1/2 - n) / (5^(1/3) * n^(3/2) * sqrt(3*Pi)). - _Vaclav Kotesovec_, Nov 28 2017
%e A120593 A(x) = 1 + x + 6*x^2 + 76*x^3 + 1201*x^4 + 21252*x^5 +...
%e A120593 A(x)^4 = 1 + 4*x + 30*x^2 + 380*x^3 + 6005*x^4 + 106260*x^5 +...
%t A120593 CoefficientList[1 + InverseSeries[Series[1+5*x - (1+x)^4, {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Nov 28 2017 *)
%o A120593 (PARI) {a(n)=local(A=1+x+6*x^2+x*O(x^n));for(i=0,n,A=A-5*A+4+x+A^4);polcoeff(A,n)}
%Y A120593 Cf. A120588 - A120592, A120594 - A120607; A244856.
%K A120593 nonn
%O A120593 0,3
%A A120593 _Paul D. Hanna_, Jun 16 2006, Jan 24 2008