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%I #12 May 10 2012 22:23:11
%S 1,1,2,9,52,372,3058,28074,282028,3059328,35497672,437499541,
%T 5696752234,78036803430,1120687989348,16823652188164,263345788211608,
%U 4289062071449610,72543038644585822,1271980596430351862,23085579883157411532,433071407705851089244
%N G.f. satisfies: A(x) = 1 + x*A(x*A(x)^2)^2.
%F Given g.f. A(x), let G(x) be defined by G(x*A(x)^2) = x, then
%F (1) G(x) = x/(1 + A(x)^2*G(x))^2,
%F (2) A(G(x)) = 1 + A(x)^2*G(x).
%e G.f. A(x) = 1 + x + 2*x^2 + 9*x^3 + 52*x^4 + 372*x^5 + 3058*x^6 +...
%e A(x)^2 = 1 + 2*x + 5*x^2 + 22*x^3 + 126*x^4 + 884*x^5 + 7149*x^6 +...
%e A(x*A(x)^2) = 1 + x + 4*x^2 + 22*x^3 + 156*x^4 + 1285*x^5 + 11886*x^6 +...
%e A(x*A(x)^2)^2 = 1 + 2*x + 9*x^2 + 52*x^3 + 372*x^4 + 3058*x^5 +...
%e Define G(x) by G(x*A(x)^2) = x, then
%e G(x) = x - 2*x^2 + 3*x^3 - 12*x^4 + 17*x^5 - 198*x^6 - 345*x^7 +...
%e such that G(x) = x/(1 + A(x)^2*G(x))^2.
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A^2,x,x*A^2));polcoeff(A,n)}
%Y Cf. A212029, A143501.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 21 2008
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