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%I #2 Mar 30 2012 18:37:10
%S 1,1,2,8,32,138,624,2922,14036,68788,342584,1728812,8820864,45428616,
%T 235846688,1232970010,6485204532,34295308230,182233431688,
%U 972493015258,5209848971700,28008206873944,151053157070944,817032258098112
%N G.f. satisfies: A(x) = x + A(A(A(A(x)^2))).
%F G.f. satisfies: A( x - A(A(A(x^2))) ) = x.
%e G.f.: A(x) = x + x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 138*x^6 + 624*x^7 +...
%e Related expansions:
%e A(A(x)) = x + 2*x^2 + 6*x^3 + 27*x^4 + 134*x^5 + 706*x^6 + 3892*x^7 +...
%e A(A(A(A(x)))) = x + 4*x^2 + 20*x^3 + 122*x^4 + 820*x^5 + 5838*x^6 +...
%e A(A(A(A(x)^2))) = x^2 + 2*x^3 + 8*x^4 + 32*x^5 + 138*x^6 + 624*x^7 +...
%e The series reversion of A(x) = x - A(A(A(x^2))), where
%e A(A(A(x^2))) = x^2 + 3*x^4 + 12*x^6 + 63*x^8 + 368*x^10 + 2282*x^12 +...
%o (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x+subst(A,x,subst(A,x,subst(A,x,A^2+x*O(x^n)))));polcoeff(A,n)}
%Y Cf. A141381, A141382, A141383; A141370.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Jun 28 2008