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A154677 G.f. satisfies: A(x/A(x)) = G(x) where G(x) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan numbers).

%I A154677 #11 Feb 04 2018 03:19:56
%S A154677 1,1,3,13,70,440,3116,24274,204407,1836339,17425275,173329307,
%T A154677 1796783304,19323703019,214843877103,2462522274426,29032815570544,
%U A154677 351447240945518,4361579736404011,55424256247911490,720399315622779670,9569215299494074698,129799982362958621827
%N A154677 G.f. satisfies: A(x/A(x)) = G(x) where G(x) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan numbers).
%H A154677 Paul D. Hanna, <a href="/A154677/b154677.txt">Table of n, a(n) for n = 0..200</a>
%F A154677 G.f. satisfies: A( (x-x^2)/A(x-x^2) ) = 1/(1-x).
%F A154677 G.f. satisfies: A( (x/(1+x)^2)/A(x/(1+x)^2) ) = 1 + x.
%F A154677 G.f. satisfies: A(x) = 1 + A(x)^2*Series_Reversion(x/A(x)). - _Paul D. Hanna_, Dec 06 2009
%e A154677 G.f.: A(x) = 1 + x + 3*x^2 + 13*x^3 + 70*x^4 + 440*x^5 + ... where
%e A154677 A(x/A(x)) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + ...
%e A154677 x/A(x) = x - x^2 - 2*x^3 - 8*x^4 - 43*x^5 - 277*x^6 - 2026*x^7 - ...
%o A154677 (PARI) {a(n)=local(A=1+x,F=sum(k=0,n,binomial(2*k+1,k)/(2*k+1)*x^k)+x*O(x^n)); for(i=0,n,A=subst(F,x,serreverse(x/(A+x*O(x^n)))));polcoeff(A,n)}
%o A154677 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+A^2*serreverse(x/(A+x*O(x^n))));polcoeff(A,n)} \\ _Paul D. Hanna_, Dec 06 2009
%Y A154677 Cf. A000108.
%Y A154677 Cf. variants: A168448, A168478. - _Paul D. Hanna_, Dec 06 2009
%K A154677 nonn
%O A154677 0,3
%A A154677 _Paul D. Hanna_, Jan 14 2009