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%I #20 Sep 03 2022 09:55:19
%S 1,0,2,3,-28,-150,2154,26040,-322512,-7872984,77570280,3752301960,
%T -22068935736,-2542757920560,1422846762960,2302464947491800,
%U 14860063644794880,-2653728770258072640,-41790782141846648640,3739260018343338345600
%N E.g.f. satisfies A(x)^(A(x)^2) = 1/(1 - x)^x.
%F a(n) = n! * Sum_{k=0..floor(n/2)} (-2*k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
%F E.g.f.: A(x) = Sum_{k>=0} (-2*k+1)^(k-1) * (-x * log(1-x))^k / k!.
%F E.g.f.: A(x) = exp( LambertW(-2 * x * log(1-x))/2 ).
%F E.g.f.: A(x) = ( -2 * x * log(1-x)/LambertW(-2 * x * log(1-x)) )^(1/2).
%o (PARI) a(n) = n!*sum(k=0, n\2, (-2*k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-2*k+1)^(k-1)*(-x*log(1-x))^k/k!)))
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-2*x*log(1-x))/2)))
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*x*log(1-x)/lambertw(-2*x*log(1-x)))^(1/2)))
%Y Cf. A355842, A356795, A356796, A356905.
%Y Cf. A349652, A356885.
%K sign
%O 0,3
%A _Seiichi Manyama_, Sep 03 2022
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