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A186632 E.g.f. satisfies: A(A(A(x))) = d/dx A(x)^2/2 with A(0)=0, A'(0)=1.

%I A186632 #8 Jul 25 2025 16:45:13
%S A186632 1,2,24,588,22560,1201320,82875744,7085373456,727827579648,
%T A186632 87810122446560,12227912455651200,1938610503725962176,
%U A186632 346053263639068417536,68919212804888587849344
%N A186632 E.g.f. satisfies: A(A(A(x))) = d/dx A(x)^2/2 with A(0)=0, A'(0)=1.
%F A186632 E.g.f. satisfies: A(A(A(x))) = A'(x)*A(x).
%F A186632 E.g.f. satisfies: A(A(A(A(x)))) = A'(A(x))*A(A(x)) = A( A'(x)*A(x) ).
%F A186632 E.g.f. satisfies: A(A(A(A(A(x))))) = A'(A(A(x)))*A'(x)*A(x).
%F A186632 n(n-1) divides a(n) for n>=2.
%e A186632 E.g.f: A(x) = x + 2*x^2/2! + 24*x^3/3! + 588*x^4/4! + 22560*x^5/5! +...
%e A186632 A(x)^2/2 = x^2/2! + 6*x^3/3! + 108*x^4/4! + 3420*x^5/5! + 158760*x^6/6! +...
%e A186632 Iterations of the e.g.f. begin:
%e A186632 A(A(x)) = x + 4*x^2/2! + 60*x^3/3! + 1680*x^4/4! + 70920*x^5/5! +...
%e A186632 A(A(A(x))) = x + 6*x^2/2! + 108*x^3/3! + 3420*x^4/4! + 158760*x^5/5! +...
%e A186632 A(A(A(A(x)))) = x + 8*x^2/2! + 168*x^3/3! + 5952*x^4/4! + 302640*x^5/5! +...
%e A186632 A(A(A(A(A(x))))) = x + 10*x^2/2! + 240*x^3/3! + 9420*x^4/4! + 522000*x^5/5! +...
%e A186632 Related expansions:
%e A186632 A'(A(x)) = 1 + 2*x + 28*x^2/2! + 780*x^3/3! + 33384*x^4/4! + 1956120*x^5/5! +...
%e A186632 A'(A(A(x))) = 1 + 2*x + 32*x^2/2! + 996*x^3/3! + 46944*x^4/4! + 2998680*x^5/5! +...
%o A186632 (PARI) {a(n)=local(A=x+x^2+sum(m=3, n-1, a(m)*x^m/m!)+x*O(x^n)); if(n<3, n!*polcoeff(A, n),n!*polcoeff(subst(A,x,subst(A,x,A))-deriv(A^2/2), n)/(n-2))}
%Y A186632 Cf. A179420.
%K A186632 nonn
%O A186632 1,2
%A A186632 _Paul D. Hanna_, Feb 24 2011