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E.g.f. satisfies A(x) = exp(x + x / A(x)).
6

%I #26 Feb 16 2025 08:34:05

%S 1,2,0,8,-64,832,-13568,269824,-6328320,171044864,-5235245056,

%T 178988498944,-6760886435840,279614956503040,-12566949343002624,

%U 609881495812702208,-31785828867471572992,1770660964785178279936,-104990165030126886060032

%N E.g.f. satisfies A(x) = exp(x + x / A(x)).

%H Seiichi Manyama, <a href="/A362693/b362693.txt">Table of n, a(n) for n = 0..372</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: x / LambertW(x*exp(-x)) = exp( x + LambertW(x*exp(-x)) ).

%F a(n) = Sum_{k=0..n} (-k+1)^(n-1) * binomial(n,k) = 2^n * A349719(n).

%t nmax = 20; A[_] = 1;

%t Do[A[x_] = Exp[x + x/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x+lambertw(x*exp(-x)))))

%Y Cf. A349562, A362694, A362734, A362735.

%Y Cf. A362736, A362737.

%Y Cf. A349719.

%K sign,changed

%O 0,2

%A _Seiichi Manyama_, May 01 2023