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%I #5 Mar 30 2012 18:37:25
%S 1,1,2,5,17,78,423,2547,16809,119633,904868,7217525,60369382,
%T 526911858,4781722888,44992996528,437927234508,4400711725541,
%U 45584253192633,486049982786691,5328493141214993,59997231748407317,693194446470892036
%N G.f.: A(x) = 1 + x*F(x)*G(x) where F(x) = A(x/F(x)) and G(x) = A(x*G(x)).
%F G.f. A(x), along with F(x) = A(x/F(x)) and G(x) = A(x*G(x)), satisfy:
%F * A(x/A(x)) = 1 + x*F(x/A(x)) since G(x/A(x)) = A(x);
%F * A(x*A(x)) = 1 + x*A(x)^2*G(x*A(x)) since F(x*A(x)) = A(x);
%F * A(x/F(x)^2) = 1 + x*F( x/F(x)^2 )/F(x) since F(x) = G(x/F(x)^2);
%F * A(x*G(x)^2) = 1 + x*G(x)^3*G( x*G(x)^2 ) since G(x) = F(x*G(x)^2).
%e G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 78*x^5 + 423*x^6 +...
%e The function F(x) = A(x/F(x)) is the g.f. of A184510 and begins:
%e F(x) = 1 + x + x^2 + x^3 + 4*x^4 + 22*x^5 + 103*x^6 + 565*x^7 + 3650*x^8 +...
%e The function G(x) = A(x*G(x)) is the g.f. of A184511 and begins:
%e G(x) = 1 + x + 3*x^2 + 12*x^3 + 58*x^4 + 324*x^5 + 2016*x^6 + 13629*x^7 + 98644*x^8 +...
%e Related expansions:
%e A(x*A(x)) = 1 + x + 3*x^2 + 11*x^3 + 47*x^4 + 236*x^5 + 1365*x^6 + 8799*x^7 + 61770*x^8 +...
%e A(x/A(x)) = 1 + x + x^2 - 2*x^4 - 2*x^5 - 4*x^6 - 55*x^7 - 281*x^8 - 1545*x^9 -...
%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*serreverse(x/A)/serreverse(x*A)+x*O(x^n));polcoeff(A,n)}
%Y Cf. A184510, A184511, A184506.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 16 2011
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