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A356902 E.g.f. satisfies A(x) * log(A(x)) = x * (exp(x) - 1).

%I A356902 #21 Feb 16 2025 08:34:03
%S A356902 1,0,2,3,-8,-55,276,4417,-13488,-639567,-248300,141842921,797525400,
%T A356902 -43103642855,-584650622724,16366430341185,436555007091616,
%U A356902 -6909610676492959,-368240758971238620,2371795171252419385,354876368637537736680,1050192150132691993161
%N A356902 E.g.f. satisfies A(x) * log(A(x)) = x * (exp(x) - 1).
%H A356902 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356902 a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.
%F A356902 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.
%F A356902 E.g.f.: A(x) = exp( LambertW(x * (exp(x) - 1)) ).
%F A356902 E.g.f.: A(x) = -x * (1 - exp(x))/LambertW(-x * (1 - exp(x))).
%t A356902 nmax = 21; A[_] = 1;
%t A356902 Do[A[x_] = Exp[(Exp[x] - 1)*x/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t A356902 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356902 (PARI) a(n) = n!*sum(k=0, n\2, (-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);
%o A356902 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(x*(exp(x)-1))^k/k!)))
%o A356902 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*(exp(x)-1)))))
%o A356902 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*(1-exp(x))/lambertw(-x*(1-exp(x)))))
%Y A356902 Cf. A355843, A356797, A356798, A356904.
%Y A356902 Cf. A349583.
%K A356902 sign
%O A356902 0,3
%A A356902 _Seiichi Manyama_, Sep 03 2022