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%I #7 Aug 14 2022 10:17:03
%S 1,0,0,1,7,43,281,2056,17004,157809,1622515,18245335,222004597,
%T 2898508416,40343356184,595578837205,9287308741827,152459628788599,
%U 2627373030049669,47425289731038656,895098852673047772,17644305594671247141,363065584549610882703,7799894520723959486795
%N a(n) = exp(-1) * n! * Sum_{k>=0} Laguerre(n,k) / k!.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LaguerrePolynomial.html">Laguerre Polynomial</a>
%F E.g.f.: exp(exp(-x/(1 - x)) - 1) / (1 - x).
%F a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * k! * Bell(n-k).
%t Table[Exp[-1] n! Sum[LaguerreL[n, k]/k!, {k, 0, Infinity}], {n, 0, 23}]
%t nmax = 23; CoefficientList[Series[Exp[Exp[-x/(1 - x)] - 1]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
%t Table[Sum[(-1)^(n - k) Binomial[n, k]^2 k! BellB[n - k], {k, 0, n}], {n, 0, 23}]
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(exp(-x/(1 - x)) - 1) / (1 - x))) \\ _Michel Marcus_, Aug 12 2022
%Y Cf. A000110, A009940, A101053, A317362, A317366.
%K nonn
%O 0,5
%A _Ilya Gutkovskiy_, Aug 12 2022