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%I #2 Mar 30 2012 18:37:22
%S 1,0,6,0,540,0,156240,0,96480720,0,104661849600,0,177947471782080,0,
%T 439942718370355200,0,1499983925770422892800,0,
%U 6780179728814814933811200,0,39412342523061395825857920000,0
%N E.g.f. satisfies: A(A(x))^3 = A(x)^3 * A'(x).
%F Let A_n(x) denote the n-th iteration of A(x), then
%F . [A_{n+1}(x)]^3 = A(x)^3 * A_n'(x) for all n,
%F and A = A(x) satisfies:
%F . A = x + A^3 + A^3*D(A^3)/2! + A^3*D(A^3*D(A^3))/3! + A^3*D(A^3*D(A^3*D(A^3)))/4! + ...;
%F . A_n(x) = x + n*A^3 + n^2*A^3*D(A^3)/2! + n^3*A^3*D(A^3*D(A^3))/3! + n^4*A^3*D(A^3*D(A^3*D(A^3)))/4! + ...
%F where operator D(F) = d/dx F.
%e E.g.f: A(x) = x + 6*x^3/3! + 540*x^5/5! + 156240*x^7/7! + 96480720*x^9/9! + 104661849600*x^11/11! + 177947471782080*x^13/13! + 439942718370355200*x^15/15! +...
%e Related expansions:
%e A(x)^3 = 6*x^3/3! + 360*x^5/5! + 83160*x^7/7! + 43908480*x^9/9! +...
%e A(A(x)) = x + 12*x^3/3! + 1440*x^5/5! + 509040*x^7/7! + 368686080*x^9/9! +...
%e A(A(x))^3 = 6*x^3/3! + 720*x^5/5! + 241920*x^7/7! + 165110400*x^9/9! +...
%e A'(x) = 1 + 6*x^2/2! + 540*x^4/4! + 156240*x^6/6! + 96480720*x^8/8! +...
%o (PARI) /* Coefficients of A_m(x) = m-th iteration of A(x): */
%o {a(n,m=1)=local(A=x+x^3,D);for(i=1,n,D=x;A=x+sum(k=1,n,m^k*(D=(A+x*O(x^n))^3*deriv(D))/k!));if(n<1,0,n!*polcoeff(A,n))}
%Y Cf. A179497 (variant).
%K eigen,nonn
%O 1,3
%A _Paul D. Hanna_, Aug 02 2010
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