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%I #17 Mar 05 2024 06:03:39
%S 1,3,-6,84,-1599,42906,-1477716,62171661,-3090518556,177237143040,
%T -11518529575857,836601742598628,-67156626492464064,
%U 5904119985344031639,-564188922815428792914,58225175660113940932032,-6453955474121138652732903,764716767229825444834522086
%N E.g.f. satisfies A(x) * log(A(x)) = 3 * (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) = Sum_{k=0..n} 3^k * (-k+1)^(k-1) * Stirling2(n,k).
%F E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (3 * (exp(x) - 1))^k / k!.
%F E.g.f.: A(x) = exp( LambertW(3 * (exp(x) - 1)) ).
%F E.g.f.: A(x) = 3 * (exp(x) - 1)/LambertW(3 * (exp(x) - 1)).
%t nmax = 17; A[_] = 1;
%t Do[A[x_] = Exp[(3*(Exp[x] - 1))/A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o (PARI) a(n) = sum(k=0, n, 3^k*(-k+1)^(k-1)*stirling(n, k, 2));
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(3*(exp(x)-1))^k/k!)))
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*(exp(x)-1)))))
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(3*(exp(x)-1)/lambertw(3*(exp(x)-1))))
%Y Cf. A349583, A357244.
%K sign
%O 0,2
%A _Seiichi Manyama_, Sep 19 2022
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