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E.g.f. A(x) = G(x)/x where G(x) satisfies: G(G(G(x))) = 2*x*G'(x) - G(x).
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%I #10 Dec 31 2012 17:48:19

%S 1,1,6,81,1828,59910,2629800,146775160,10047085200,821599116300,

%T 78674552192800,8684916065005620,1091429676788178240,

%U 154543476785542516360,24445478524707259098240,4288239906998845117572000,829048705765475214447735040

%N E.g.f. A(x) = G(x)/x where G(x) satisfies: G(G(G(x))) = 2*x*G'(x) - G(x).

%H Paul D. Hanna, <a href="/A193265/b193265.txt">Table of n, a(n) for n = 0..149</a>

%F a(n) = A193264(n+1)/(n+1).

%e E.g.f.: A(x) = 1 + x + 6*x^2/2! + 81*x^3/3! + 1828*x^4/4! + 59910*x^5/5! +...

%e Let G(x) = x*A(x), then:

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

%e which equals 2*x*G'(x) - G(x) = x*A(x) + 2*x^2*A'(x).

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

%Y Cf. A193264.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 20 2011