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%I #11 Nov 20 2017 03:24:39
%S 1,1,2,15,104,1445,18144,364651,6761600,176898249,4376614400,
%T 140703601511,4370369292288,166520945009965,6235421191430144,
%U 274675339364551875,12046634866183798784,602474837696641959569,30289753591657339944960,1696072847731424941183039
%N Expansion of e.g.f. 2/(1 + sqrt(1 - 4*LambertW(x))).
%H Robert Israel, <a href="/A295268/b295268.txt">Table of n, a(n) for n = 0..379</a>
%F E.g.f.: 1/(1 - LambertW(x)/(1 - LambertW(x)/(1 - LambertW(x)/(1 - LambertW(x)/(1 - ...))))), a continued fraction.
%F a(n) ~ 2^(2*n + 3/2) * n^(n-1) / (sqrt(5) * exp(5*n/4)). - _Vaclav Kotesovec_, Nov 19 2017
%p S:= series(2/(1 + sqrt(1 - 4*LambertW(x))), x, 31):
%p seq(coeff(S,x,n)*n!,n=0..30); # _Robert Israel_, Nov 20 2017
%t nmax = 19; CoefficientList[Series[2/(1 + Sqrt[1 - 4 LambertW[x]]), {x, 0, nmax}], x] Range[0, nmax]!
%t nmax = 19; CoefficientList[Series[1/(1 + ContinuedFractionK[-LambertW[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!
%o (PARI) x = 'x + O('x^30); Vec(serlaplace(2/(1 + sqrt(1 - 4*lambertw(x))))) \\ _Michel Marcus_, Nov 20 2017
%Y Cf. A000108, A180680, A277184, A295267.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 19 2017
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