%I #11 Feb 25 2014 02:18:41

%S 1,1,1,4,12,56,310,1872,13804,110368,990792,9816560,105392056,

%T 1231910208,15473322592,208287327136,2992281160320,45647837225984,

%U 737580584547424,12578608722516480,225799744451927104

%N E.g.f. satisfies: A'(x) = 1 + x*A(x)^2 where A(0) = 1.

%H Vincenzo Librandi, <a href="/A144012/b144012.txt">Table of n, a(n) for n = 0..200</a>

%F E.g.f. satisfies: A(x) = 1 + Integral [1 + x*A(x)^2] dx.

%F Let r be the radius of convergence of e.g.f. A(x), then: a(n)/n! ~ r^(n+2) where r=0.89757966985304971385345783421642045642527022484..., A(-r)=0.206876159989240..., A'(-r)=0.961585613659124...

%F r is the root of the equation 2*r^2*Hypergeometric0F1[1/3,-1/(9*r^3)] = Hypergeometric0F1[5/3,-1/(9*r^3)]. - _Vaclav Kotesovec_, Feb 23 2014

%e E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 12*x^4/4! + 56*x^5/5! +...

%e A(x)^2 = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 62*x^4/4! + 312*x^5/5! +...

%e x*A(x)^2 = x + 4*x^2/2! + 12*x^3/3! + 56*x^4/4! + 310*x^5/5! +...

%e A'(x) = 1 + x + 4*x^2/2! + 12*x^3/3! + 56*x^4/4! + 310*x^5/5! +...

%t CoefficientList[Series[-2*(Hypergeometric0F1[2/3,-x^3/9] + x*Hypergeometric0F1[4/3,-x^3/9]) / (-2*Hypergeometric0F1[1/3,-x^3/9] + x^2*Hypergeometric0F1[5/3,-x^3/9]),{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Dec 21 2013 *)

%o (PARI) {a(n)=local(A=1+x); for(i=0, n, A=1+intformal(1+x*(A+x*O(x^n))^2)); n!*polcoeff(A, n)}

%Y Cf. A144013, A144014.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Sep 10 2008