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A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.

%I A349583 #30 Feb 16 2025 08:34:02
%S A349583 1,1,0,2,-9,72,-710,8563,-121814,1997502,-37097739,769687954,
%T A349583 -17644355410,442894514285,-12081668234012,355889274553166,
%U A349583 -11258683640579857,380701046875217492,-13702507978018209458,523049811008797507683,-21105565578064063658754
%N A349583 E.g.f. satisfies: A(x) * log(A(x)) = exp(x) - 1.
%H A349583 Seiichi Manyama, <a href="/A349583/b349583.txt">Table of n, a(n) for n = 0..399</a>
%H A349583 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349583 a(n) = Sum_{k=0..n} (-k+1)^(k-1) * Stirling2(n,k).
%F A349583 E.g.f.: A(x) = exp( LambertW(exp(x) - 1) ).
%F A349583 G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
%F A349583 a(n) ~ -(-1)^n * sqrt(exp(1) - 1) * n^(n-1) / (exp(n+1) * (1 - log(exp(1) - 1))^(n - 1/2)). - _Vaclav Kotesovec_, Dec 05 2021
%p A349583 b:= proc(n, m) option remember; `if`(n=0,
%p A349583      (1-m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
%p A349583     end:
%p A349583 a:= n-> b(n, 0):
%p A349583 seq(a(n), n=0..24);  # _Alois P. Heinz_, Jul 29 2022
%t A349583 a[n_] := Sum[If[k == 1, 1, (-k + 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 21, 0] (* _Amiram Eldar_, Nov 23 2021 *)
%o A349583 (PARI) a(n) = sum(k=0, n, (-k+1)^(k-1)*stirling(n, k, 2));
%o A349583 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(exp(x)-1))))
%o A349583 (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 A349583 Cf. A120980, A349561, A349585.
%Y A349583 Cf. A008277, A052880, A349524, A349525.
%K A349583 sign
%O A349583 0,4
%A A349583 _Seiichi Manyama_, Nov 22 2021