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E.g.f. satisfies A(x) * log(A(x)) = x * (exp(x) - 1).
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%I #20 Mar 04 2024 11:53:39

%S 1,0,2,3,-8,-55,276,4417,-13488,-639567,-248300,141842921,797525400,

%T -43103642855,-584650622724,16366430341185,436555007091616,

%U -6909610676492959,-368240758971238620,2371795171252419385,354876368637537736680,1050192150132691993161

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * Stirling2(n-k,k)/(n-k)!.

%F E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (x * (exp(x) - 1))^k / k!.

%F E.g.f.: A(x) = exp( LambertW(x * (exp(x) - 1)) ).

%F E.g.f.: A(x) = -x * (1 - exp(x))/LambertW(-x * (1 - exp(x))).

%t nmax = 21; A[_] = 1;

%t Do[A[x_] = Exp[(Exp[x] - 1)*x/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)

%o (PARI) a(n) = n!*sum(k=0, n\2, (-k+1)^(k-1)*stirling(n-k, k, 2)/(n-k)!);

%o (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 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(x*(exp(x)-1)))))

%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*(1-exp(x))/lambertw(-x*(1-exp(x)))))

%Y Cf. A355843, A356797, A356798, A356904.

%Y Cf. A349583.

%K sign

%O 0,3

%A _Seiichi Manyama_, Sep 03 2022