%I #6 Mar 26 2016 06:55:47
%S 1,5,33,713,259985,33800019393,571220656468785748897,
%T 163146519188315271151004136501803467012673,
%U 13308393361631661480558768899208178264126697372663086894845840270509196618153817473,88556666933960837614967248701566245757216163828988828731024437766134512448872086575494410071109970379702252158611451372093516192879655317859248726893259500181148929
%N G.f.: 1/(1-x) = sqrt(1 + a(1)*x + x*sqrt(1 + a(2)*x + x*sqrt(1 + a(3)*x + x*sqrt(...) ) ) ).
%C Compare g.f. to: 1+x = sqrt(1+x + x*sqrt(1+x + x*sqrt(1+x + x*sqrt(...)))).
%C Limit a(n+1)/a(n)^2 = 1/2.
%F a(n) ~ 2 * c^(2^n), where c = 1.4448099826529... . - _Vaclav Kotesovec_, Mar 26 2016
%e G.f.: 1/(1-x) = sqrt(1 + x + x*sqrt(1 + 5*x + x*sqrt(1 + 33*x + x*sqrt(1 + 713*x + x*sqrt(1 + 259985*x + x*sqrt(1 + 33800019393*x + x*sqrt(1 + 571220656468785748897*x + x*sqrt(1 +...)))))))).
%e Illustrate the calculation of initial terms.
%e 1: 1/(1-x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6 +...
%e 2: ((1/(1-x)^2 - 1 - x)/x)^2 = 1 + 6*x + 17*x^2 + 34*x^3 + 58*x^4 + 90*x^5 + 131*x^6 +...
%e 3: (( ((1/(1-x)^2 - 1 - x)/x)^2 - 1 - 5*x)/x)^2 = 1 + 34*x + 357*x^2 + 1272*x^3 + 3308*x^4 + 7266*x^5 + 14302*x^6 +...
%e 4: (( (( ((1/(1-x)^2 - 1 - x)/x)^2 - 1 - 5*x)/x)^2 - 1 - 33*x)/x)^2 = 1 + 714*x + 129993*x^2 + 914824*x^3 + 3994428*x^4 + 13632080*x^5 + 39691244*x^6 +...
%e 5: (( (( (( ((1/(1-x)^2 - 1 - x)/x)^2 - 1 - 5*x)/x)^2 - 1 - 33*x)/x)^2 - 1 - 713*x)/x)^2 = 1 + 259986*x + 16900009697*x^2 + 237849421320*x^3 + 1875425573144*x^4 + 10852626534712*x^5 +...
%t poly = 1/(1-x)^2; Table[coef = SeriesCoefficient[poly, {x, 0, 1}] - 1; poly = ((poly - 1 - coef*x)/x)^2; coef, {n, 1, 10}] (* _Vaclav Kotesovec_, Mar 26 2016 *)
%o (PARI) {a(n) = my(F=1/(1-x +x^2*O(x^n))); for(i=1,n-1, F = ((F^2 - 1+x - polcoeff(F^2,1)*x)/x)); polcoeff(F^2,1)-1}
%o for(n=1,12,print1(a(n),", "))
%K nonn
%O 1,2
%A _Paul D. Hanna_, Mar 26 2016