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A234289 E.g.f. satisfies: A(x) = 1 + A(x)^2 * Integral 1/A(x) dx.

%I A234289 #20 May 23 2019 19:07:24
%S A234289 1,1,3,17,147,1729,25827,468593,10012083,246287521,6856204803,
%T A234289 213102768977,7315460977107,274894137157249,11223280473993507,
%U A234289 494715928976218673,23416019742035332083,1184519963466363339361,63774753426394808946243,3641219528568659379843857
%N A234289 E.g.f. satisfies: A(x) = 1 + A(x)^2 * Integral 1/A(x) dx.
%C A234289 Compare to: G(x) = 1 + G(x)^2 * Integral 1/G(x)^2 dx, where G(x) is the e.g.f. of A006351, the number of series-parallel networks with n labeled edges.
%H A234289 Paul D. Hanna, <a href="/A234289/b234289.txt">Table of n, a(n) for n = 0..200</a>
%F A234289 E.g.f.: 1 / ( d/dx Series_Reversion( Integral C(x) dx ) ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108.
%F A234289 E.g.f.: 1 + Series_Reversion( 2*x/(1+x) - log(1+x) ).
%F A234289 E.g.f.: -2/LambertW(-1,-2*exp(x-2)). - _Vaclav Kotesovec_, Dec 27 2013
%F A234289 E.g.f.: A(x) = C( Integral 1/A(x) dx ), where C(x) = 1 + x*C(x)^2 = (1 - sqrt(1-4*x))/(2*x), is the Catalan function of A000108. - _Paul D. Hanna_, May 23 2019
%F A234289 a(n) ~ 2 * n^(n-1) / (exp(n) * (1-log(2))^(n-1/2)). - _Vaclav Kotesovec_, Dec 27 2013
%e A234289 E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 147*x^4/4! + 1729*x^5/5! +...
%e A234289 where A(x)^2 = 1 + 2*x + 8*x^2/2! + 52*x^3/3! + 484*x^4/4! + 5948*x^5/5! +...
%e A234289 Integral 1/A(x) dx = x - x^2/2! - x^3/3! - 5*x^4/4! - 41*x^5/5! - 469*x^6/6! +...
%e A234289 Further,
%e A234289 Series_Reversion(Integral 1/A(x) dx) = x + x^2/2 + 2*x^3/3 + 5*x^4/4 + 14*x^5/5 + 42*x^6/6 + 132*x^7/7 +...+ A000108(n-1)*x^n/n +...
%e A234289 where A000108(n) = binomial(2*n,n)/(n+1).
%p A234289 seq(n! * coeff(series(-2/LambertW(-1,-2*exp(x-2)), x, n+1), x, n), n = 0..10) # _Vaclav Kotesovec_, Dec 27 2013
%t A234289 CoefficientList[1 + InverseSeries[Series[2*x/(1+x) - Log[1+x], {x, 0, 20}], x],x]* Range[0, 20]! (* _Vaclav Kotesovec_, Dec 27 2013 *)
%o A234289 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+A^2*intformal(1/(A+x*O(x^n))));n!*polcoeff(A,n)}
%o A234289 for(n=0,25,print1(a(n),", "))
%o A234289 (PARI) /* Explicit formula using Catalan function C(x) = 1 + x*C(x)^2: */
%o A234289 {a(n)=local(C=(1-sqrt(1-4*x+x^2*O(x^n)))/(2*x),A=1); A=1/deriv(serreverse(intformal(C))); n!*polcoeff(A, n)}
%o A234289 for(n=0, 25, print1(a(n), ", "))
%o A234289 (PARI) /* Explicit formula: 1 + Series_Reversion(2*x/(1+x) - log(1+x)): */
%o A234289 {a(n)=local(A=1,X=x+x^2*O(x^n)); A=1+serreverse(2*X/(1+X)-log(1+X)); n!*polcoeff(A, n)}
%o A234289 for(n=0, 25, print1(a(n), ", "))
%Y A234289 Cf. A000108, A234290.
%K A234289 nonn
%O A234289 0,3
%A A234289 _Paul D. Hanna_, Dec 22 2013