A376106 Expansion of e.g.f. LambertW(x / (1 - 2*x)).
0, 1, 2, 9, 56, 465, 4764, 58345, 830192, 13466817, 245254580, 4955259441, 109995693576, 2661003245329, 69682488950060, 1963774182830265, 59261538449833184, 1906643335934717697, 65149411890671521380, 2356212733788818122561, 89920484394446094721400
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..400
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Mathematica
nmax=20; CoefficientList[InverseSeries[Series[x / (2*x + E^(-x)), {x, 0, nmax}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 20 2024 *) -
PARI
my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(lambertw(x/(1-2*x))))) -
PARI
a(n) = n!*sum(k=1, n, 2^(n-k)*(-k)^(k-1)*binomial(n-1, k-1)/k!);
Formula
E.g.f. A(x) satisfies A(x) = x * (2*A(x) + exp(-A(x))).
E.g.f.: Series_Reversion( x / (2*x + exp(-x)) ).
a(n) = n! * Sum_{k=1..n} 2^(n-k) * (-k)^(k-1) * binomial(n-1,k-1)/k!.