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A349585 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).

%I A349585 #30 Feb 16 2025 08:34:02
%S A349585 1,1,-2,8,-59,642,-9112,158839,-3279880,78250188,-2117569181,
%T A349585 64082989720,-2144319848772,78609355884893,-3133061858717806,
%U A349585 134884905211588892,-6238095343894356675,308427209934965151158,-16234730389499986865092,906409067599064528054343
%N A349585 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).
%H A349585 Seiichi Manyama, <a href="/A349585/b349585.txt">Table of n, a(n) for n = 0..378</a>
%H A349585 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349585 a(n) = (-1)^(n-1) * Sum_{k=0..n} (k-1)^(k-1) * Stirling2(n,k).
%F A349585 E.g.f.: A(x) = exp( LambertW(1 - exp(-x)) ).
%F A349585 G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
%F A349585 a(n) ~ -(-1)^n * sqrt(1 + exp(1)) * n^(n-1) / (exp(n+1) * (log(1 + exp(1)) - 1)^(n - 1/2)). - _Vaclav Kotesovec_, Dec 05 2021
%p A349585 b:= proc(n, m) option remember; `if`(n=0,
%p A349585      (m-1)^(m-1), m*b(n-1, m)+b(n-1, m+1))
%p A349585     end:
%p A349585 a:= n-> (-1)^(n-1)*b(n, 0):
%p A349585 seq(a(n), n=0..20);  # _Alois P. Heinz_, Aug 03 2022
%t A349585 a[n_] := (-1)^(n - 1) * Sum[If[k == 1, 1, (k - 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 19, 0] (* _Amiram Eldar_, Nov 23 2021 *)
%o A349585 (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (k-1)^(k-1)*stirling(n, k, 2));
%o A349585 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(1-exp(-x)))))
%o A349585 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))
%Y A349585 Cf. A120980, A349561, A349583.
%Y A349585 Cf. A008277, A058864, A349527, A349528.
%K A349585 sign
%O A349585 0,3
%A A349585 _Seiichi Manyama_, Nov 22 2021