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G.f. satisfies: A(x) = B(x*A(x)), where B(x) is the g.f. of A184509.
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%I #8 Mar 30 2012 18:37:25

%S 1,1,3,12,58,324,2016,13629,98644,756852,6110309,51620412,454430088,

%T 4155005770,39354004740,385288338532,3892135131803,40507984800374,

%U 433792913778315,4774455016668509,53954983308058733,625485598856053837,7432389116043114682

%N G.f. satisfies: A(x) = B(x*A(x)), where B(x) is the g.f. of A184509.

%F G.f. B(x) of A184509 satisfies: B(x) = 1 + x*A(x)*F(x) where F(x) = B(x/F(x)) = A(x/F(x)^2) is the g.f. of A184510 and A(x) is the g.f. of this sequence.

%e G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 58*x^4 + 324*x^5 + 2016*x^6 +...

%e where A(x) = B(x*A(x)) and B(x) = A(x/B(x)) is the g.f. of A184509:

%e B(x) = 1 + x + 2*x^2 + 5*x^3 + 17*x^4 + 78*x^5 + 423*x^6 + 2547*x^7 +...

%e Also, A(x) = F(x*A(x)^2) where F(x) = A(x/F(x)^2) is the g.f. of A184510:

%e F(x) = 1 + x + x^2 + x^3 + 4*x^4 + 22*x^5 + 103*x^6 + 565*x^7 +...

%e The product A(x)*F(x) begins:

%e A(x)*F(x) = 1 + 2*x + 5*x^2 + 17*x^3 + 78*x^4 + 423*x^5 + 2547*x^6 +...

%e where B(x) = 1 + x*A(x)*F(x).

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

%Y Cf. A184509, A184510.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 13 2011