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A120566 G.f. satisfies: A(x) = A(A(x)) - x*A(A(A(x))), with A(0)=0.

%I A120566 #7 Apr 30 2014 01:32:53
%S A120566 1,1,1,3,7,33,109,643,2623,17929,85349,652395,3517911,29484193,
%T A120566 176844781,1605009651,10575269935,103033059513,738834271605,
%U A120566 7676696689275,59466011617671,655467253898577,5451048833933693
%N A120566 G.f. satisfies: A(x) = A(A(x)) - x*A(A(A(x))), with A(0)=0.
%C A120566 If A(0, x) = x, A(n+1, x) = A( A(n, x)) = A(n, A(x)). Then A(n, x) = x + n*x^2 + n^2*x^3 + (n^3 + 2*n)*x^4 + (n^4 + 6*n^2)*x^5 + ... where [x^4] A(n, x) = A054602(n). - _Michael Somos_, Jan 22 2012
%F A120566 G.f. satisfies: A(-A(-x)) = x ; Also: A(x) = x + A(A(x))*series_reversion(A(x)).
%F A120566 Since g.f. satisfies: A(A(x)) = ( x - A(x) )/A(-x), then higher order self-compositions of A(x) reduce into expressions involving A(x) and A(-x). - _Paul D. Hanna_, Jul 22 2006
%e A120566 A(x) = x + x^2 + x^3 + 3x^4 + 7x^5 + 33x^6 + 109x^7 + 643x^8 +...
%e A120566 A(A(x)) = x + 2x^2 + 4x^3 + 12x^4 + 40x^5 + 168x^6 + 736x^7 + 3784x^8+..
%e A120566 x*A(A(A(x))) = x^2 + 3x^3 + 9x^4 + 33x^5 + 135x^6 + 627x^7 + 3141x^8+...
%o A120566 (PARI) {a(n)=local(A=x+x^2+x*O(x^n));if(n<1,0, for(i=1,n,A=x-subst(A,x,-x)*subst(A,x,A));polcoeff(A,n))}
%K A120566 nonn
%O A120566 1,4
%A A120566 _Paul D. Hanna_, Jun 14 2006