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%I #13 Nov 06 2019 04:21:47
%S 1,1,3,24,181,1893,20601,245176,3018669,38198478,493218343,6441378129,
%T 84807054552,1120545910725,14820493111536,195812569428897,
%U 2580287366558579,33878771120862777,443012040333754728,5770422757461475027,74931929672784252306
%N G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^9)^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 (A000108).
%C (3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 (A001764).
%C (4) E(x) = 1 + x/E(-x*E(x)^7)^4 when E(x) = 1 + x*E(x)^4 (A002293).
%C (5) F(x) = 1 + x/F(-x*F(x)^9)^5 when F(x) = 1 + x*F(x)^5 (A002294).
%C The first negative term is a(68). - _Georg Fischer_, Feb 16 2019
%H Paul D. Hanna, <a href="/A213100/b213100.txt">Table of n, a(n) for n = 0..300</a>
%e G.f.: A(x) = 1 + x + 3*x^2 + 24*x^3 + 181*x^4 + 1893*x^5 + 20601*x^6 +...
%e Related expansions:
%e A(x)^9 = 1 + 9*x + 63*x^2 + 516*x^3 + 4563*x^4 + 45207*x^5 + 486579*x^6 +...
%e A(-x*A(x)^9)^3 = 1 - 3*x - 15*x^2 - 64*x^3 - 798*x^4 - 8277*x^5 - 99411*x^6 -...
%t m = 21; A[_] = 1; Do[A[x_] = 1 + x/A[-x A[x]^9]^3 + O[x]^m, {m}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 06 2019 *)
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x/subst(A^3,x,-x*subst(A^9,x,x+x*O(x^n))) );polcoeff(A,n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A000108, A001764, A002293, A002294, A213091, A213092, A213093, A213094, A213095, A213096, A213098, A213099, A213101, A213102, A213103, A213104, A213105.
%K sign
%O 0,3
%A _Paul D. Hanna_, Jun 05 2012
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