A295267 Expansion of e.g.f. 2/(1 + sqrt(1 + 4*LambertW(-x))).

1, 1, 6, 63, 952, 18885, 465696, 13764667, 475039104, 18767660553, 835805555200, 41442148754391, 2264776308946944, 135268340058044557, 8767315076546568192, 612911076907734961875, 45973645939542007054336, 3683096368557198711874833, 313878687736263437290438656

Offset: 0

Author

Ilya Gutkovskiy, Nov 19 2017

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..354

Crossrefs

Cf. A000108, A180680, A277184, A295268.

Programs

Maple

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

Mathematica

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

PARI

x ='x+O('x^30); Vec(serlaplace(2/(1 +sqrt(1 +4*lambertw(-x))))) \\ G. C. Greubel, Jul 07 2018

Formula

E.g.f.: 1/(1 + LambertW(-x)/(1 + LambertW(-x)/(1 + LambertW(-x)/(1 + LambertW(-x)/(1 + ...))))), a continued fraction.

a(n) ~ 2^(2*n + 3/2) * n^(n-1) / (sqrt(3) * exp(3*n/4)). - Vaclav Kotesovec, Nov 19 2017