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Expansion of e.g.f. 2/(1 + sqrt(1 + 4*LambertW(-x))).
2

%I #15 Mar 27 2019 10:02:41

%S 1,1,6,63,952,18885,465696,13764667,475039104,18767660553,

%T 835805555200,41442148754391,2264776308946944,135268340058044557,

%U 8767315076546568192,612911076907734961875,45973645939542007054336,3683096368557198711874833,313878687736263437290438656

%N Expansion of e.g.f. 2/(1 + sqrt(1 + 4*LambertW(-x))).

%H G. C. Greubel, <a href="/A295267/b295267.txt">Table of n, a(n) for n = 0..354</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(3) * exp(3*n/4)). - _Vaclav Kotesovec_, Nov 19 2017

%p a:=series(2/(1+sqrt(1+4*LambertW(-x))),x=0,19): seq(n!*coeff(a,x,n),n=0..18); # _Paolo P. Lava_, Mar 27 2019

%t nmax = 18; CoefficientList[Series[2/(1 + Sqrt[1 + 4 LambertW[-x]]), {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 18; 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))))) \\ _G. C. Greubel_, Jul 07 2018

%Y Cf. A000108, A180680, A277184, A295268.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 19 2017