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A213093 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^4).

%I A213093 #14 Nov 01 2019 18:36:51
%S A213093 1,1,1,4,13,62,297,1523,8091,43243,234347,1267141,6814076,36368431,
%T A213093 192079140,1006805203,5262612068,27656507707,147973596219,
%U A213093 815825605806,4662818005761,27504894986209,165036600363916,989160502170958,5829789341752240,33444482725193880
%N A213093 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^4).
%C A213093 Compare definition of g.f. to:
%C A213093 (1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
%C A213093 (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 A213093 (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 A213093 The first negative term is a(42) = -16825305705383790675462237694. - _Georg Fischer_, Feb 16 2019
%H A213093 Paul D. Hanna, <a href="/A213093/b213093.txt">Table of n, a(n) for n = 0..300</a>
%e A213093 G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 13*x^4 + 62*x^5 + 297*x^6 + 1523*x^7 +...
%e A213093 Related expansions:
%e A213093 A(x)^4 = 1 + 4*x + 10*x^2 + 32*x^3 + 119*x^4 + 516*x^5 + 2462*x^6 +...
%e A213093 A(-x*A(x)^4) = 1 - x - 3*x^2 - 6*x^3 - 31*x^4 - 141*x^5 - 697*x^6 - 3641*x^7 -...
%t A213093 nmax = 25; sol = {a[0] -> 1};
%t A213093 Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x/A[-x A[x]^4]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t A213093 sol /. Rule -> Set;
%t A213093 a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 01 2019 *)
%o A213093 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A,x,-x*subst(A^4,x,x+x*O(x^n))) );polcoeff(A,n)}
%o A213093 for(n=0,30,print1(a(n),", "))
%Y A213093 Cf. A000108, A001764, A213091, A213092, A213094, A213095, A213096.
%K A213093 sign
%O A213093 0,4
%A A213093 _Paul D. Hanna_, Jun 05 2012