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%I #19 Sep 16 2023 15:20:07
%S 0,-1,1,1,2,6,24,120,720,5040,40320,362880,3628800,39916800,479001600,
%T 6227020800,87178291200,1307674368000,20922789888000,355687428096000,
%U 6402373705728000,121645100408832000,2432902008176640000,51090942171709440000,1124000727777607680000,25852016738884976640000
%N Coefficients of x^n/n! in the expansion of (1 - x)*log(1 - x).
%C The negated sequence gives the compositional inverse of 1 - exp(W(-x)) with the principal branch of Lambert's W function.
%C See A000312 for the two formulas of _Vladimir Kruchinin_, Sep 15 2010, and _Peter Bala_, Dec 09 2011. Here stated the other way around.
%H R. J. Mathar, <a href="https://arxiv.org/abs/0803.0900">Series of reciprocal powers of k-almost primes</a>, arXiv:0803.0900 [math.NT], (2018) eq (60).
%F E.g.f.: (1 - x) * log(1 - x).
%F a(0) = 0, a(1) = -1, a(n) = (n-2)! = A000142(n-2), for n >= 2.
%t With[{m = 25}, Range[0, m]! * CoefficientList[Series[(1-x) * Log[1-x], {x, 0, m}], x]] (* _Amiram Eldar_, Nov 14 2022 *)
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)*log(1-x)), -N) \\ _Michel Marcus_, Sep 16 2023
%Y Cf. A000142, A103505, A000312, A159333, A104150.
%K sign,easy
%O 0,5
%A _Wolfdieter Lang_, Nov 14 2022