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A213099 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^7)^3.

%I A213099 #13 Nov 06 2019 04:22:08
%S A213099 1,1,3,18,112,909,7833,74603,740541,7656219,81187518,878435208,
%T A213099 9647220024,107137240686,1199914011387,13521738420240,153051832116378,
%U A213099 1737562815056865,19762347822563532,224970273310192579,2561375647064514444,29149168085832027732
%N A213099 G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^7)^3.
%C A213099 Compare definition of g.f. to:
%C A213099 (1) B(x) = 1 + x/B(-x*B(x)) when B(x) = 1/(1-x).
%C A213099 (2) C(x) = 1 + x/C(-x*C(x)^3)^2 when C(x) = 1 + x*C(x)^2 (A000108).
%C A213099 (3) D(x) = 1 + x/D(-x*D(x)^5)^3 when D(x) = 1 + x*D(x)^3 (A001764).
%C A213099 (4) E(x) = 1 + x/E(-x*E(x)^7)^4 when E(x) = 1 + x*E(x)^4 (A002293).
%C A213099 (5) F(x) = 1 + x/F(-x*F(x)^9)^5 when F(x) = 1 + x*F(x)^5 (A002294).
%C A213099 The first negative term is a(121). - _Georg Fischer_, Feb 16 2019
%H A213099 Paul D. Hanna, <a href="/A213099/b213099.txt">Table of n, a(n) for n = 0..300</a>
%e A213099 G.f.: A(x) = 1 + x + 3*x^2 + 18*x^3 + 112*x^4 + 909*x^5 + 7833*x^6 +...
%e A213099 Related expansions:
%e A213099 A(x)^7 = 1 + 7*x + 42*x^2 + 287*x^3 + 2079*x^4 + 16611*x^5 + 142702*x^6 +...
%e A213099 A(-x*A(x)^7)^3 = 1 - 3*x - 9*x^2 - 31*x^3 - 318*x^4 - 2586*x^5 - 25969*x^6 -...
%t A213099 m = 22; A[_] = 1; Do[A[x_] = 1 + x/A[-x A[x]^7]^3 + O[x]^m, {m}];
%t A213099 CoefficientList[A[x], x] (* _Jean-François Alcover_, Nov 06 2019 *)
%o A213099 (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^7,x,x+x*O(x^n))) );polcoeff(A,n)}
%o A213099 for(n=0,30,print1(a(n),", "))
%Y A213099 Cf. A000108, A001764, A002293, A002294, A213091, A213092, A213093, A213094, A213095, A213096, A213098, A213100, A213101, A213102, A213103, A213104, A213105.
%K A213099 sign
%O A213099 0,3
%A A213099 _Paul D. Hanna_, Jun 05 2012