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G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^9)^4.
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%I #13 Nov 06 2019 04:22:21

%S 1,1,4,30,240,2433,26388,315726,3958452,51863952,698988716,9637772716,

%T 135161761860,1920878419569,27583547221596,399310273694328,

%U 5817100622299116,85152975761167179,1251046169511714720,18428780031111768466,271964652432415737596

%N G.f. satisfies: A(x) = 1 + x/A(-x*A(x)^9)^4.

%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(142). - _Georg Fischer_, Feb 16 2019

%H Paul D. Hanna, <a href="/A213102/b213102.txt">Table of n, a(n) for n = 0..300</a>

%e G.f.: A(x) = 1 + x + 4*x^2 + 30*x^3 + 240*x^4 + 2433*x^5 + 26388*x^6 +...

%e Related expansions:

%e A(x)^9 = 1 + 9*x + 72*x^2 + 642*x^3 + 6030*x^4 + 61551*x^5 + 670344*x^6 +...

%e A(-x*A(x)^9)^4 = 1 - 4*x - 14*x^2 - 64*x^3 - 797*x^4 - 8188*x^5 - 104090*x^6 -...

%t m = 21; A[_] = 1; Do[A[x_] = 1 + x/A[-x A[x]^9]^4 + 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^4,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, A213100, A213101, A213103, A213104, A213105.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jun 05 2012