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G.f. satisfies: A(x - x*A(x) - x*A(x)^2) = x.
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%I #5 Mar 30 2012 18:37:29

%S 1,1,4,21,134,968,7662,65135,587040,5559342,54965230,564651110,

%T 6004908296,65920345700,745289233564,8661959227407,103330815828292,

%U 1263608418272768,15823268263301680,202712359166886406,2654710188935753950,35514167158635839770

%N G.f. satisfies: A(x - x*A(x) - x*A(x)^2) = x.

%C Compare to g.f. C(x) of the Catalan numbers: C(x - x*C(x) + x*C(x)^2) = x.

%e G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 134*x^5 + 968*x^6 + 7662*x^7 +...

%e Related expansions.

%e A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 50*x^5 + 326*x^6 + 2372*x^7 + 18773*x^8 +...

%e A(x) + A(x)^2 = x + 2*x^2 + 6*x^3 + 30*x^4 + 184*x^5 + 1294*x^6 +...

%e where the series reversion of A(x) begins:

%e x-x*A(x)-x*A(x)^2 = x - x^2 - 2*x^3 - 6*x^4 - 30*x^5 - 184*x^6 - 1294*x^7 - 10034*x^8 +...

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

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 18 2011