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A356926 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^exp(x).

%I A356926 #22 Feb 16 2025 08:34:03
%S A356926 1,1,2,3,10,35,121,1092,5216,39321,558643,2433508,48144944,688652549,
%T A356926 2176310995,145742587616,1334993574032,5551320939809,799648465754835,
%U A356926 1049695714507276,90069170433616208,6281942689646504501,-53282051261767839293,2356158301117802408472
%N A356926 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^exp(x).
%H A356926 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356926 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-exp(x) * log(1-x))^k / k!.
%F A356926 E.g.f.: A(x) = exp( LambertW(-exp(x) * log(1-x)) ).
%F A356926 E.g.f.: A(x) = -exp(x) * log(1-x)/LambertW(-exp(x) * log(1-x)).
%t A356926 nmax = 23; A[_] = 1;
%t A356926 Do[A[x_] = ((1 - x)^(-Exp[x]))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356926 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356926 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-exp(x)*log(1-x))^k/k!)))
%o A356926 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-exp(x)*log(1-x)))))
%o A356926 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*log(1-x)/lambertw(-exp(x)*log(1-x))))
%Y A356926 Cf. A191365, A356925.
%Y A356926 Cf. A356912, A356913.
%Y A356926 Cf. A002104, A177885.
%Y A356926 Cf. A141209.
%K A356926 sign
%O A356926 0,3
%A A356926 _Seiichi Manyama_, Sep 04 2022