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%I #16 Nov 01 2019 18:36:45
%S 1,1,1,3,8,31,120,511,2234,9988,45497,208435,959496,4414091,20252947,
%T 92586100,421351615,1910531192,8647504950,39194735661,178643040883,
%U 822295086652,3836023988259,18167435295220,87268076036356,423657019406289,2067868784722846
%N G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^3).
%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(54) = -4736158402689145255029229896601957. - _Georg Fischer_, Feb 16 2019
%H Paul D. Hanna, <a href="/A213092/b213092.txt">Table of n, a(n) for n = 0..300</a>
%e G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 8*x^4 + 31*x^5 + 120*x^6 + 511*x^7 +...
%e Related expansions:
%e A(x)^3 = 1 + 3*x + 6*x^2 + 16*x^3 + 48*x^4 + 171*x^5 + 664*x^6 + 2760*x^7 +...
%e A(-x*A(x)^3) = 1 - x - 2*x^2 - 3*x^3 - 14*x^4 - 50*x^5 - 213*x^6 - 915*x^7 -...
%t nmax = 26; 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]^3]) + 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^3,x,x+x*O(x^n))) );polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A000108, A001764, A213091, A213093, A213094, A213095, A213096.
%K sign
%O 0,4
%A _Paul D. Hanna_, Jun 05 2012
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