%I #9 Dec 31 2012 17:44:58

%S 1,2,18,324,9140,359460,18408600,1174201280,90423766800,8215991163000,

%T 865420074120800,104218992780067440,14188585798246317120,

%U 2163608674997595229040,366682177870608886473600,68611838511981521881152000,14093827998013078645611495680

%N E.g.f. A(x) satisfies: A(A(A(x))) = 2*x*A'(x) - A(x), where A(x) = Sum_{n>=1} a(n)*x^n, with a(1)=1, a(2)=2.

%H Paul D. Hanna, <a href="/A193264/b193264.txt">Table of n, a(n) for n = 1..150</a>

%F E.g.f. A(x) satisfies: A(A(A(x))) = Sum_{n>=1} (2*n-1)*a(n)*x^n.

%F a(n) = n*A193265(n-1).

%e E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 324*x^4/4! + 9140*x^5/5! + 359460*x^6/6! +...+ a(n)*x^n/n! +...

%e where

%e A(A(A(x))) = x + 6*x^2/2! + 90*x^3/3! + 2268*x^4/4! + 82260*x^5/5! + 3954060*x^6/6! +...+ (2*n-1)*a(n)*x^n/n! +...

%e which equals:

%e 2*x*A'(x) - A(x) = x + 3*2*x^2/2! + 5*18*x^3/3! + 7*324*x^4/4! + 9*9140*x^5/5! +...

%o (PARI) {a(n)=local(A=x);if(n<1,0,if(n<=2,n,A=x+sum(m=2,n-1,a(m)*x^m/m!)+x*O(x^n);

%o n!*polcoeff(subst(A,x,subst(A,x,A))-2*x*A',n)/(2*n-4)))}

%Y Cf. A193265, A179420.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 20 2011