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

%I #17 Mar 04 2024 08:47:23

%S 1,1,-2,6,-52,540,-7608,129304,-2612608,60867360,-1608663840,

%T 47527158624,-1552431588288,55547889458880,-2160724031160576,

%U 90782738645280000,-4097139872604807168,197675862365363088384,-10153243488783257091072

%N E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(1 - x).

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

%F E.g.f. satisfies A(x)^A(x) * (1 - x)^(1 - x) = 1.

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

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

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

%t nmax = 20; A[_] = 1;

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

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

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-(1-x)*log(1-x))^k/k!)))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-(1-x)*log(1-x)))))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-(1-x)*log(1-x)/lambertw(-(1-x)*log(1-x))))

%Y Cf. A005727, A155456, A349561, A356905, A356908.

%K sign

%O 0,3

%A _Seiichi Manyama_, Sep 19 2022