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A259267 E.g.f. A(x) satisfies: A'(x) = exp(2*A(A(x))).

%I A259267 #6 Jul 04 2015 22:56:45
%S A259267 1,2,12,128,2016,42656,1145280,37563008,1464675840,66533778944,
%T A259267 3466031815680,204489094565888,13524452573872128,994257291909816320,
%U A259267 80668058806271016960,7179145234347383128064,697131195162680465817600,73522035747248454761578496,8387016414085244676889116672
%N A259267 E.g.f. A(x) satisfies: A'(x) = exp(2*A(A(x))).
%F A259267 E.g.f. A(x) satisfies:
%F A259267 (1) A''(x) = 2*exp( 4*A(A(x)) + 2*A(A(A(x))) ).
%F A259267 (2) exp(-2*A(x)) = d/dx Series_Reversion(A(x)).
%F A259267 (3) A(x) = log(F(x)) where F(x) satisfies: F( Integral 1/F(x)^2 dx ) = exp(x) and equals the e.g.f. of A233336.
%F A259267 a(n) = 2^(n-1) * A214645(n) for n>=1.
%e A259267 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 2016*x^5/5! +...
%e A259267 Related expansions:
%e A259267 A'(x) =  1 + 2*x + 12*x^2/2! + 128*x^3/3! + 2016*x^4/4! + 42656*x^5/5! +...
%e A259267 A(A(x)) = log(A'(x))/2 = x + 4*x^2/2! + 36*x^3/3! + 520*x^4/4! + 10512*x^5/5! + 276064*x^6/6! + 8987712*x^7/7! + 351278080*x^8/8! +...
%e A259267 The exponential of e.g.f. A(x) equals the e.g.f. of A233336:
%e A259267 exp(A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 201*x^4/4! + 3097*x^5/5! + 63963*x^6/6! + 1677883*x^7/7! +...+ A233336(n)*x^n/n! +...
%o A259267 (PARI) {a(n)=local(A=x+x^2); for(i=0, n, A=intformal(exp(2*subst(A, x, A+x*O(x^n))))); n!*polcoeff(A, n)}
%o A259267 for(n=1, 25, print1(a(n), ", "))
%Y A259267 Cf. A233336, A214645.
%K A259267 nonn
%O A259267 1,2
%A A259267 _Paul D. Hanna_, Jul 03 2015