cp's OEIS Frontend

A357243 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(1 - x).

%I A357243 #18 Feb 16 2025 08:34:04
%S A357243 1,1,-2,6,-52,540,-7608,129304,-2612608,60867360,-1608663840,
%T A357243 47527158624,-1552431588288,55547889458880,-2160724031160576,
%U A357243 90782738645280000,-4097139872604807168,197675862365363088384,-10153243488783257091072
%N A357243 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(1 - x).
%H A357243 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A357243 E.g.f. satisfies A(x)^A(x) * (1 - x)^(1 - x) = 1.
%F A357243 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-(1-x) * log(1-x))^k / k!.
%F A357243 E.g.f.: A(x) = exp( LambertW(-(1-x) * log(1-x)) ).
%F A357243 E.g.f.: A(x) = -(1-x) * log(1-x)/LambertW(-(1-x) * log(1-x)).
%t A357243 nmax = 20; A[_] = 1;
%t A357243 Do[A[x_] = ((1 - x)^(-1 + x))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A357243 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A357243 (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 A357243 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-(1-x)*log(1-x)))))
%o A357243 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-(1-x)*log(1-x)/lambertw(-(1-x)*log(1-x))))
%Y A357243 Cf. A005727, A155456, A349561, A356905, A356908.
%K A357243 sign
%O A357243 0,3
%A A357243 _Seiichi Manyama_, Sep 19 2022