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A356905 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.

%I A356905 #23 Feb 16 2025 08:34:03
%S A356905 1,0,2,3,-4,-30,294,3780,-7904,-444528,78840,99657360,539299848,
%T A356905 -27852945120,-361237078944,10124338180320,258341121976320,
%U A356905 -4020500134465920,-205187357182405824,1330097523844832640,186823640933648588160,500469438126120583680
%N A356905 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^x.
%H A356905 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356905 a(n) = n! * Sum_{k=0..floor(n/2)} (-k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.
%F A356905 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x * log(1-x))^k / k!.
%F A356905 E.g.f.: A(x) = exp( LambertW(-x * log(1-x)) ).
%F A356905 E.g.f.: A(x) = -x * log(1-x)/LambertW(-x * log(1-x)).
%t A356905 nmax = 21; A[_] = 1;
%t A356905 Do[A[x_] = (1/(1 - x)^x)^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356905 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356905 (PARI) a(n) = n!*sum(k=0, n\2, (-k+1)^(k-1)*abs(stirling(n-k, k, 1))/(n-k)!);
%o A356905 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x*log(1-x))^k/k!)))
%o A356905 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x*log(1-x)))))
%o A356905 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-x*log(1-x)/lambertw(-x*log(1-x))))
%Y A356905 Cf. A355842, A356795, A356796, A356906.
%Y A356905 Cf. A349561, A356884.
%K A356905 sign
%O A356905 0,3
%A A356905 _Seiichi Manyama_, Sep 03 2022