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%I #15 Nov 01 2019 18:36:39
%S 1,1,1,2,4,11,31,98,317,1070,3685,12928,45924,164552,593398,2148288,
%T 7796846,28328601,102948125,373955584,1357252616,4921292287,
%U 17828236695,64546901169,233660589210,846258569786,3068523234989,11147449003438,40600425590874,148330067463010
%N G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^2).
%C Compare definition of g.f. to:
%C (1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
%C (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 (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 The first negative term is a(80). - _Georg Fischer_, Feb 16 2019
%H Paul D. Hanna, <a href="/A213091/b213091.txt">Table of n, a(n) for n = 0..300</a>
%e G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 11*x^5 + 31*x^6 + 98*x^7 +...
%e Related expansions:
%e A(x)^2 = 1 + 2*x + 3*x^2 + 6*x^3 + 13*x^4 + 34*x^5 + 96*x^6 + 296*x^7 +...
%e A(-x*A(x)^2) = 1 - x - x^2 - x^3 - 4*x^4 - 10*x^5 - 34*x^6 - 107*x^7 -...
%t nmax = 29; sol = {a[0] -> 1};
%t Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - (1 + x/A[(-x) A[x]^2]) + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
%t sol /. Rule -> Set;
%t a /@ Range[0, nmax] (* _Jean-François Alcover_, Nov 01 2019 *)
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A,x,-x*subst(A^2,x,x+x*O(x^n))) );polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A000108, A001764, A213092, A213093, A213094, A213095, A213096.
%K sign
%O 0,4
%A _Paul D. Hanna_, Jun 05 2012