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%I #3 Sep 03 2012 09:23:37
%S 1,1,3,21,181,1815,20154,241665,3080331,41302128,578300961,8410381731,
%T 126533164302,1963198249559,31334634890994,513482316151767,
%U 8625106767201627,148306563373055094,2607509241485053311,46832500213908831048,858557964325898228058
%N G.f. A(x) satisfies: A( x - A(x)^3 ) = x+x^3 - A(x)^3.
%F G.f. A(x) satisfies: A(x) = x + G(x)^3 where G(x - A(x)^3) = x.
%e G.f.: A(x) = x + x^3 + 3*x^5 + 21*x^7 + 181*x^9 + 1815*x^11 + 20154*x^13 +...
%e Related expansions:
%e A(x)^3 = x^3 + 3*x^5 + 12*x^7 + 82*x^9 + 705*x^11 + 6999*x^13 + 76881*x^15 +...
%e A(x-A(x)^3) = x - 3*x^5 - 12*x^7 - 82*x^9 - 705*x^11 - 6999*x^13 -...
%e x+x^3 - A(x)^3 = x - 3*x^5 - 12*x^7 - 82*x^9 - 705*x^11 - 6999*x^13 -...
%e Let G(x) equal the series reversion of x - A(x)^3:
%e G(x) = x + x^3 + 6*x^5 + 48*x^7 + 467*x^9 + 5124*x^11 + 61284*x^13 + 783129*x^15 +...
%e then
%e G(x)^3 = x^3 + 3*x^5 + 21*x^7 + 181*x^9 + 1815*x^11 + 20154*x^13 +...
%e A(G(x)) = x + 2*x^3 + 12*x^5 + 105*x^7 + 1096*x^9 + 12816*x^11 +...
%e A(G(x))^3 = x^3 + 6*x^5 + 48*x^7 + 467*x^9 + 5124*x^11 + 61284*x^13 +...
%e where A(x) = x + G(x)^3 = G(x) + G(x)^3 - A(G(x))^3.
%o (PARI) {a(n)=local(A=x+x^2);for(i=1,n,A=x+serreverse(x-A^3+x*O(x^n))^3);polcoeff(A,n)}
%o for(n=1,30,print1(a(2*n-1),", ")) \* only odd-indexed terms are shown *\
%Y Cf. A214404.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Sep 03 2012
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