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A356909 E.g.f. satisfies A(x)^A(x) = 1/(1 - 2*x)^(1/2).

%I A356909 #14 Feb 16 2025 08:34:03
%S A356909 1,1,1,6,25,260,2031,29414,343729,6331464,99347775,2213854962,
%T A356909 43775364777,1143045179484,27318538754703,817488701386590,
%U A356909 22930078960458081,773538392955105552,24914603847636688383,935548306987015373418,34024565857537814925465
%N A356909 E.g.f. satisfies A(x)^A(x) = 1/(1 - 2*x)^(1/2).
%H A356909 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A356909 a(n) = Sum_{k=0..n} 2^(n-k) * (-k+1)^(k-1) * |Stirling1(n,k)|.
%F A356909 E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-log(1-2*x)/2)^k / k!.
%F A356909 E.g.f.: A(x) = exp( LambertW(-log(1-2*x)/2) ).
%F A356909 E.g.f.: A(x) = -log(1-2*x)/(2 * LambertW(-log(1-2*x)/2)).
%t A356909 nmax = 20; A[_] = 1;
%t A356909 Do[A[x_] = (1 - 2*x)^(-(1/2)/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
%t A356909 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A356909 (PARI) a(n) = sum(k=0, n, 2^(n-k)*(-k+1)^(k-1)*abs(stirling(n, k, 1)));
%o A356909 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-log(1-2*x)/2)^k/k!)))
%o A356909 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-log(1-2*x)/2))))
%o A356909 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(-log(1-2*x)/(2*lambertw(-log(1-2*x)/2))))
%Y A356909 Cf. A355786, A356908.
%K A356909 nonn
%O A356909 0,4
%A A356909 _Seiichi Manyama_, Sep 03 2022