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A356908 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^2.

%I A356908 #19 Feb 16 2025 08:34:03
%S A356908 1,2,-2,24,-272,4840,-107496,2934400,-94501760,3511914624,
%T A356908 -147882696960,6959217277056,-361941871760256,20616017227643136,
%U A356908 -1276341094954066176,85337532623368181760,-6128269375791673718784,470426144496265208979456
%N A356908 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^2.
%H A356908 Robert Israel, <a href="/A356908/b356908.txt">Table of n, a(n) for n = 0..350</a>
%H A356908 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356908 a(n) = Sum_{k=0..n} 2^k * (-k+1)^(k-1) * |Stirling1(n,k)|.
%F A356908 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-2 * log(1-x))^k / k!.
%F A356908 E.g.f.: A(x) = exp( LambertW(-2 * log(1-x)) ).
%F A356908 E.g.f.: A(x) = -2 * log(1-x)/LambertW(-2 * log(1-x)).
%p A356908 S:= series(exp( LambertW(-2 * log(1-x))),x,51):
%p A356908 seq(n!*coeff(S,x,n),n=0..50); # _Robert Israel_, Sep 18 2022
%t A356908 nmax = 20; A[_] = 1;
%t A356908 Do[A[x_] = (1/(1-x)^2)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356908 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o A356908 (PARI) a(n) = sum(k=0, n, 2^k*(-k+1)^(k-1)*abs(stirling(n, k, 1)));
%o A356908 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-2*log(1-x))^k/k!)))
%o A356908 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-2*log(1-x)))))
%o A356908 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-2*log(1-x)/lambertw(-2*log(1-x))))
%Y A356908 Cf. A349561, A355779, A356909.
%K A356908 sign
%O A356908 0,2
%A A356908 _Seiichi Manyama_, Sep 03 2022