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A232690 E.g.f. satisfies: A(x) = exp( 1/A(x) * Integral A(x)^3 dx ).

%I A232690 #18 Jan 05 2014 17:09:58
%S A232690 1,1,2,7,33,202,1495,13107,132062,1508629,19227687,270818542,
%T A232690 4173948097,69906444393,1263811926338,24534217063999,508951297964193,
%U A232690 11236656534791578,263054502440239639,6508910392250017611,169727899004807970782,4652123984505282141277,133711980572082349859559
%N A232690 E.g.f. satisfies: A(x) = exp( 1/A(x) * Integral A(x)^3 dx ).
%C A232690 Note that G(x) = exp(1/G(x) * Integral G(x)^2 dx) has negative coefficients.
%C A232690 Compare e.g.f. to: B(x) = exp( 1/B(x) * Integral B(x) dx ) where B(y) = Bessel polynomial y_n(-1) (cf. A000806).
%H A232690 Vaclav Kotesovec, <a href="/A232690/b232690.txt">Table of n, a(n) for n = 0..100</a>
%F A232690 E.g.f.: sqrt(LambertW(-1,(4*x-3)*exp(-3))/(4*x-3)). - _Vaclav Kotesovec_, Jan 05 2014
%F A232690 Limit n->infinity (a(n)/n!)^(1/n) = 4/3. - _Vaclav Kotesovec_, Jan 05 2014
%e A232690 E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 33*x^4/4! + 202*x^5/5! +...
%e A232690 Related expansions:
%e A232690 log(A(x)) = x + x^2/2! + 3*x^3/3! + 11*x^4/4! + 61*x^5/5! + 393*x^6/6! +...
%e A232690 Integral A(x)^3 dx = x + 3*x^2/2! + 12*x^3/3! + 63*x^4/4! + 411*x^5/5! +...
%e A232690 1/A(x) = 1 - x - x^3/3! - x^4/4! - 12*x^5/5! - 41*x^6/6! - 451*x^7/7! -...
%p A232690 seq(n! * coeff(series(sqrt(LambertW(-1,(4*x-3)*exp(-3))/(4*x-3)), x, n+1), x, n), n=0..20); # _Vaclav Kotesovec_, Jan 05 2014
%t A232690 CoefficientList[FullSimplify[Assuming[Element[x, Reals], Series[Sqrt[LambertW[-1,(4*x-3)*E^(-3)]/(4*x-3)], {x, 0, 20}]]], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 05 2014 *)
%o A232690 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(1/A*intformal(A^3+x*O(x^n))));n!*polcoeff(A,n)}
%o A232690 for(n=0,30,print1(a(n),", "))
%Y A232690 Cf. A232691, A232692.
%K A232690 nonn
%O A232690 0,3
%A A232690 _Paul D. Hanna_, Dec 06 2013