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A192948 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^5 - 1)^n.

%I A192948 #12 Jan 28 2017 21:30:49
%S A192948 1,1,5,60,885,14605,258126,4778340,91460415,1795377600,35946770255,
%T A192948 731245323256,15070729457030,314011160190675,6603561278126200,
%U A192948 139980599432879480,2987856960226960551,64162892863813071450,1385270621375211268550
%N A192948 G.f. satisfies: A(x) = 1 + x*Sum_{n>=0} (A(x)^5 - 1)^n.
%C A192948 Compare to a g.f. of the Catalan numbers: C(x) = 1 + x*Sum_{n>=0} (C(x) - 1)^n.
%H A192948 G. C. Greubel, <a href="/A192948/b192948.txt">Table of n, a(n) for n = 0..725</a>
%H A192948 Vaclav Kotesovec, <a href="/A192948/a192948.txt">Recurrence</a>
%F A192948 G.f. A(x) equals the formal inverse of function (x-1)*(2-x^5).
%F A192948 G.f. satisfies: A(x) = 1 + x/(2 - A(x)^5).
%F A192948 G.f.: A(x) = 1 + Series_Reversion( 2*x - x*(1+x)^5 ).
%F A192948 a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 23.55179049164234... is the root of the equation -46656 - 385675*d - 1217000*d^2 - 1775000*d^3 - 1100000*d^4 + 50000*d^5 = 0 and c = 0.03694932533778987590415707531... - _Vaclav Kotesovec_, Sep 17 2013
%e A192948 G.f.: A(x) = 1 + x + 5*x^2 + 60*x^3 + 885*x^4 + 14605*x^5 +...
%e A192948 where (A(x) - 1)*(2 - A(x)^5) = x
%e A192948 and A(x - 5*x^2 - 10*x^3 - 10*x^4 - 5*x^5 - x^6) = 1 + x.
%e A192948 Related expansions.
%e A192948 (A(x)^5-1) = 5*x + 35*x^2 + 410*x^3 + 6030*x^4 + 99376*x^5 +...
%e A192948 (A(x)^5-1)^2 = 25*x^2 + 350*x^3 + 5325*x^4 + 89000*x^5 +...
%e A192948 (A(x)^5-1)^3 = 125*x^3 + 2625*x^4 + 49125*x^5 + 925625*x^6 +...
%e A192948 (A(x)^5-1)^4 = 625*x^4 + 17500*x^5 + 388750*x^6 + 8177500*x^7 +...
%t A192948 CoefficientList[1+InverseSeries[Series[2*x-x*(1+x)^5,{x,0,20}],x],x] (* _Vaclav Kotesovec_, Sep 17 2013 *)
%o A192948 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x*sum(m=0,n,(A^5-1+x*O(x^n))^m));polcoeff(A,n)}
%o A192948 (PARI) {a(n)=local(A=1+serreverse(2*x-x*(1+x)^5+x^2*O(x^n)));polcoeff(A,n)}
%Y A192948 Cf. A192945, A192946, A192947.
%K A192948 nonn
%O A192948 0,3
%A A192948 _Paul D. Hanna_, Jul 13 2011