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A357246 E.g.f. satisfies A(x) * log(A(x)) = (1-x) * (exp(x) - 1).

%I A357246 #19 Feb 16 2025 08:34:04
%S A357246 1,1,-2,5,-49,497,-6926,116510,-2325422,53538315,-1397740279,
%T A357246 40792008435,-1316056239994,46509292766172,-1786748828967402,
%U A357246 74139054468535061,-3304409577659864305,157444695280699565069,-7986085592316390890618,429645521271113815480246
%N A357246 E.g.f. satisfies A(x) * log(A(x)) = (1-x) * (exp(x) - 1).
%H A357246 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A357246 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * ((1-x) * (exp(x) - 1))^k / k!.
%F A357246 E.g.f.: A(x) = exp( LambertW((1-x) * (exp(x) - 1)) ).
%F A357246 E.g.f.: A(x) = (1-x) * (exp(x) - 1)/LambertW((1-x) * (exp(x) - 1)).
%t A357246 nmax = 19; A[_] = 1;
%t A357246 Do[A[x_] = Exp[-(((Exp[x]-1)*(x-1))/A[x])]+O[x]^(nmax+1)//Normal, {nmax}];
%t A357246 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 05 2024 *)
%o A357246 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*((1-x)*(exp(x)-1))^k/k!)))
%o A357246 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw((1-x)*(exp(x)-1)))))
%o A357246 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((1-x)*(exp(x)-1)/lambertw((1-x)*(exp(x)-1))))
%Y A357246 Cf. A356902, A357243, A357247.
%K A357246 sign
%O A357246 0,3
%A A357246 _Seiichi Manyama_, Sep 19 2022