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%I #15 Aug 26 2023 19:45:16
%S 1,-1,4,7,120,1373,21028,373931,7670736,178064281,4615519884,
%T 132139421423,4141235867992,141016013784917,5184372688776180,
%U 204668397165154867,8635388122600110240,387787185320578895537,18467131524896950511644
%N Expansion of e.g.f. exp(-2*x) / (1 + LambertW(-x)).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F G.f.: Sum_{k>=0} (k*x)^k / (1 + 2*x)^(k+1).
%F a(n) = Sum_{k=0..n} (-2)^(n-k) * k^k * binomial(n,k).
%t With[{nn=20},CoefficientList[Series[Exp[-2x]/(1+LambertW[-x]),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Aug 26 2023 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1 + lambertw(-x))))
%Y Column k=2 of A362856.
%Y Cf. A362859.
%K sign
%O 0,3
%A _Seiichi Manyama_, May 05 2023
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