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A213095 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^5)^2.

%I A213095 #12 Nov 05 2019 11:50:50
%S A213095 1,1,2,9,40,242,1528,10664,76956,575245,4395910,34131621,268146598,
%T A213095 2122399923,16884293154,134689290877,1075641369024,8588548510081,
%U A213095 68496446989330,545303352881863,4331918361300882,34337864000400360,271657823631727330,2146133623039711577
%N A213095 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^5)^2.
%C A213095 Compare definition of g.f. to:
%C A213095 (1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
%C A213095 (2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).
%C A213095 (3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 is the g.f. of the ternary tree numbers (A001764).
%C A213095 The first negative term is a(85). - _Georg Fischer_, Feb 16 2019
%H A213095 Paul D. Hanna, <a href="/A213095/b213095.txt">Table of n, a(n) for n = 0..300</a>
%e A213095 G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 26*x^4 + 123*x^5 + 622*x^6 + 3490*x^7 +...
%e A213095 Related expansions:
%e A213095 A(x)^5 = 1 + 5*x + 20*x^2 + 95*x^3 + 485*x^4 + 2801*x^5 + 17560*x^6 +...
%e A213095 A(-x*A(x)^5)^2 = 1 - 2*x - 5*x^2 - 12*x^3 - 93*x^4 - 550*x^5 - 3981*x^6 -...
%t A213095 m = 23; A[_] = 1; Do[A[x_] = 1 + x/A[-x A[x]^5 + O[x]^m]^2 // Normal, {m}];
%t A213095 CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 05 2019 *)
%o A213095 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^2,x,-x*subst(A^5,x,x+x*O(x^n))) );polcoeff(A,n)}
%o A213095 for(n=0,30,print1(a(n),", "))
%Y A213095 Cf. A000108, A001764, A213091, A213092, A213093, A213094, A213096.
%K A213095 sign
%O A213095 0,3
%A A213095 _Paul D. Hanna_, Jun 05 2012