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A228608 E.g.f. A(x) satisfies: A'(x) = A(x)^2 + A(x)^4.

%I A228608 #21 Jan 26 2014 08:50:49
%S A228608 1,2,12,128,1968,39488,977088,28742912,979744512,37968868352,
%T A228608 1648597834752,79272057049088,4181485522464768,240067201819885568,
%U A228608 14902137637759008768,994529776192394166272,71009035425186633940992,5401058272888913168433152,435991257271370763778916352
%N A228608 E.g.f. A(x) satisfies: A'(x) = A(x)^2 + A(x)^4.
%H A228608 Vaclav Kotesovec, <a href="/A228608/b228608.txt">Table of n, a(n) for n = 0..350</a>
%F A228608 E.g.f. A(x) satisfies:
%F A228608 (1) A(x) = exp( Integral A(x) + A(x)^3 dx ) with A(0)=1.
%F A228608 (2) A(x) = (1 + B(x))/(1 - B(x)) where B(x) = tan(1-x - 1/A(x)).
%F A228608 (3) log(A(x)) = Series_Reversion( 1-exp(-x) - atan(tanh(x/2)) ).
%F A228608 (4) A( 1-exp(-x) - atan(tanh(x/2)) ) = exp(x).
%F A228608 a(n) ~ n! / (GAMMA(1/3) * 3^(1/3) * n^(2/3) * (1-Pi/4)^(n+1/3)). - _Vaclav Kotesovec_, Jan 26 2014
%e A228608 E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 128*x^3/3! + 1968*x^4/4! + 39488*x^5/5! +...
%e A228608 Related expansions.
%e A228608 A(x)^2 = 1 + 4*x + 32*x^2/2! + 400*x^3/3! + 6848*x^4/4! + 149056*x^5/5! +...
%e A228608 A(x)^4 = 1 + 8*x + 96*x^2/2! + 1568*x^3/3! + 32640*x^4/4! + 828032*x^5/5! +...
%e A228608 The logarithm of e.g.f. A(x) begins:
%e A228608 log(A(x)) = 2*x + 8*x^2/2! + 72*x^3/3! + 992*x^4/4! + 18336*x^5/5! +...
%e A228608 and equals Integral A(x) + A(x)^3 dx, where
%e A228608 A(x)^3 = 1 + 6*x + 60*x^2/2! + 864*x^3/3! + 16368*x^4/4! + 385344*x^5/5! +...
%t A228608 CoefficientList[Exp[InverseSeries[Series[1-Exp[-x]-ArcTan[Tanh[x/2]], {x, 0, 20}], x]],x]*Range[0, 20]! (* _Vaclav Kotesovec_, Dec 20 2013 *)
%o A228608 (PARI) /* Explicit formula: */
%o A228608 {a(n)=local(A,X=x+x^2*O(x^n));A=exp(serreverse(1-exp(-X) - atan(tanh(X/2))));n!*polcoeff(A,n)}
%o A228608 for(n=0,20,print1(a(n),", "))
%o A228608 (PARI) /* By definition: A'(x) = A(x)^2 + A(x)^4: */
%o A228608 {a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A^2+A^4+x*O(x^n))); n!*polcoeff(A, n)}
%o A228608 for(n=0,20,print1(a(n),", "))
%o A228608 (PARI) /* From: A(x) = exp( Integral A(x) + A(x)^3 dx ): */
%o A228608 {a(n)=local(A=1+x); for(i=1, n, A=exp(intformal(A+A^3)+x*O(x^n))); n!*polcoeff(A, n)}
%o A228608 for(n=0,20,print1(a(n),", "))
%Y A228608 Cf. A112487, A124214.
%K A228608 nonn
%O A228608 0,2
%A A228608 _Paul D. Hanna_, Dec 18 2013