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A144682 E.g.f. satisfies: A(x/A(x)^2) = exp(x).

%I A144682 #7 Mar 30 2012 18:37:14
%S A144682 1,1,5,73,2073,92481,5775133,471058953,47961475313,5904863932609,
%T A144682 858738633997941,144899744540718729,27970301202134146441,
%U A144682 6106540658691499524993,1493749158085983126737165
%N A144682 E.g.f. satisfies: A(x/A(x)^2) = exp(x).
%F A144682 E.g.f. satisfies: A(x) = exp( x*A(log A(x))^2 ).
%F A144682 E.g.f. satisfies: a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2).
%F A144682 E.g.f. A(x) = G(x/2)^2 where G(x/G(x)) = exp(x) and G(x) is the g.f. of A144681.
%e A144682 E.g.f. A(x) = 1 + x + 5*x^2/2! + 73*x^3/3! + 2073*x^4/4! +...
%e A144682 ILLUSTRATE FORMULA a(n+1) = [x^n/n!] exp(x)*A(x)^(2*n+2) as follows.
%e A144682 Form a table of coefficients of x^k/k! in exp(x)*A(x)^(2*n) for n>=1, k>=0:
%e A144682 exp(x)*A(x)^2: [(1), 3, 17, 219, 5665, 239283, 14432433, ...];
%e A144682 exp(x)*A(x)^4: [1,(5), 41, 605, 15633, 638325, 37250233, ...];
%e A144682 exp(x)*A(x)^6: [1, 7,(73), 1207, 31825, 1274407, 72322201, ...];
%e A144682 exp(x)*A(x)^8: [1, 9, 113,(2073), 56545, 2249769, 124959057, ...];
%e A144682 exp(x)*A(x)^10:[1, 11, 161, 3251,(92481), 3695451, 202282081, ...];
%e A144682 exp(x)*A(x)^12:[1, 13, 217, 4789, 142705,(5775133), 313637833, ...];
%e A144682 exp(x)*A(x)^14:[1, 15, 281, 6735, 210673, 8688975,(471058953), ...]; ...
%e A144682 then the terms along the main diagonal form this sequence shift left.
%o A144682 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,A=exp(serreverse(x/A^2)));n!*polcoeff(A,n)}
%o A144682 {a(n)=local(A=1+x+sum(k=2, n-1, a(k)*x^k/k!)+x*O(x^n)); if(n==0, 1, (n-1)!*polcoeff(exp(x+x*O(x^n))*A^(2*n), n-1))}
%Y A144682 Cf. A144681, A144683, A144684.
%K A144682 nonn
%O A144682 0,3
%A A144682 _Paul D. Hanna_, Sep 19 2008