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G.f. satisfies: x = A(x - A^2(x - A^3(x - A^4(x - A^5(x -...))))).
2

%I #7 Sep 06 2013 21:33:37

%S 1,1,4,22,150,1173,10148,94925,945972,9941558,109382042,1253180343,

%T 14889508598,182880186850,2316222358948,30188514546787,

%U 404233232273788,5553341502305868,78182001588059148,1126836634877469526,16612944202665079800,250345193073480290120

%N G.f. satisfies: x = A(x - A^2(x - A^3(x - A^4(x - A^5(x -...))))).

%H Paul D. Hanna, <a href="/A228883/b228883.txt">Table of n, a(n) for n = 1..150</a>

%e G.f.: A(X) = x + x^2 + 4*x^3 + 22*x^4 + 150*x^5 + 1173*x^6 + 10148*x^7 +...

%e Let G(x) be the series reversion of A(x), then

%e B(x)^2 = x - G(x) = x^2 + 2*x^3 + 7*x^4 + 40*x^5 + 277*x^6 + 2200*x^7 + 19211*x^8 +...

%e B(x) = x + x^2 + 3*x^3 + 17*x^4 + 117*x^5 + 932*x^6 + 8178*x^7 + 77396*x^8 +...

%e C(x)^3 = x - G(B(x)) = x^3 + 3*x^4 + 15*x^5 + 88*x^6 + 618*x^7 + 4872*x^8 +...

%e C(x) = x + x^2 + 4*x^3 + 21*x^4 + 144*x^5 + 1131*x^6 + 9828*x^7 + 92275*x^8 +...

%e D(x)^4 = x - G(C(x)) = x^4 + 4*x^5 + 22*x^6 + 140*x^7 + 1005*x^8 + 7980*x^9 +...

%e D(x) = x + x^2 + 4*x^3 + 22*x^4 + 149*x^5 + 1166*x^6 + 10096*x^7 + 94517*x^8 +...

%e E(x)^5 = x - G(D(x)) = x^5 + 5*x^6 + 30*x^7 + 200*x^8 + 1475*x^9 + 11841*x^10 +...

%e E(x) = x + x^2 + 4*x^3 + 22*x^4 + 150*x^5 + 1172*x^6 + 10140*x^7 + 94862*x^8 +...

%e ...

%o (PARI) {a(n)=local(A=x+x^2,G=x^(n+1));for(i=1,n+1,A=serreverse(x-G+x^2*O(x^n));G=x^(n+1)+x^2*O(x^n);for(k=0,n-1,G=subst(A^(n-k+1),x,x-G+x^2*O(x^n))));polcoeff(A,n)}

%o for(n=1,25,print1(a(n),", "))

%Y Cf. A228862.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 06 2013