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%I #21 Nov 27 2017 18:18:01
%S 1,1,7,95,1614,30718,626434,13383650,295692145,6700461777,
%T 154871912815,3637093846055,86539594779772,2081721640140460,
%U 50542732376144460,1236960716959913020,30483096737455969766,755783491624380578998,18839297079646725396450
%N G.f. satisfies: A(x) = (4 + A(x)^4) / (5-x).
%F G.f. satisfies:
%F (1) A(x) = 1 + Series_Reversion( (1+5*x - (1+x)^4)/(1 + x) ).
%F (2) A(x) = Sum_{n>=0} C(4*n,n)/(3*n+1) * (4 + x*A(x))^(3*n+1) / 5^(4*n+1).
%F (3) A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where G(x) = (4+x + G(x)^4)/5 is the g.f. of A120593.
%F a(n) ~ 2^(n/2 - 2) * 3^(3*(n-1)/4) / (sqrt(Pi) * n^(3/2) * (5*sqrt(2)*3^(3/4) - 16)^(n - 1/2)). - _Vaclav Kotesovec_, Nov 27 2017
%e G.f.: A(x) = 1 + x + 7*x^2 + 95*x^3 + 1614*x^4 + 30718*x^5 +...
%e Compare A(x)^4 to (5-x)*A(x):
%e A(x)^4 = 1 + 4*x + 34*x^2 + 468*x^3 + 7975*x^4 + 151976*x^5 +...
%e (5-x)*A(x) = 5 + 4*x + 34*x^2 + 468*x^3 + 7975*x^4 + 151976*x^5 +...
%t CoefficientList[1 + InverseSeries[Series[(1+5*x - (1+x)^4)/(1+x), {x, 0, 20}], x], x] (* _Vaclav Kotesovec_, Nov 27 2017 *)
%o (PARI) {a(n)=polcoeff(1 + serreverse((1+5*x - (1+x)^4)/(1 + x +x*O(x^n))), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n)=local(A=[1], Ax=1+x); for(i=1, n, A=concat(A, 0); Ax=Ser(A); A[#A]=Vec( ( Ax^4 - (5-x)*Ax ) )[#A]); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A120593, A244627, A244594.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jul 09 2014
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