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A349652 E.g.f. satisfies: A(x)^(A(x)^2) = 1/(1 - x).

%I A349652 #27 Feb 16 2025 08:34:02
%S A349652 1,1,-2,18,-220,3880,-86868,2376836,-76735216,2856604464,
%T A349652 -120457684320,5675047644288,-295430737430112,16840861797433440,
%U A349652 -1043322313406139648,69798144929293516800,-5014888682767294232832,385130588783629323238656
%N A349652 E.g.f. satisfies: A(x)^(A(x)^2) = 1/(1 - x).
%H A349652 Seiichi Manyama, <a href="/A349652/b349652.txt">Table of n, a(n) for n = 0..356</a>
%H A349652 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349652 a(n) = (-1)^(n-1) * Sum_{k=0..n} (2*k-1)^(k-1) * Stirling1(n,k).
%F A349652 E.g.f. A(x) = -Sum_{k>=0} (2*k-1)^(k-1) * (log(1-x))^k / k!.
%F A349652 E.g.f.: A(x) = ( -2*log(1-x)/LambertW(-2*log(1-x)) )^(1/2).
%F A349652 a(n) ~ -(-1)^n * n^(n-1) / (sqrt(2) * exp(n + exp(-1)/4) * (-1 + exp(exp(-1)/2))^(n - 1/2)). - _Vaclav Kotesovec_, Nov 24 2021
%t A349652 nmax = 20; A[_] = 1;
%t A349652 Do[A[x_] = (1/(1 - x))^(1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
%t A349652 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A349652 (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (2*k-1)^(k-1)*stirling(n, k, 1));
%o A349652 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (2*k-1)^(k-1)*log(1-x)^k/k!)))
%o A349652 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((-2*log(1-x)/lambertw(-2*log(1-x)))^(1/2)))
%Y A349652 Cf. A349650, A349654, A349656.
%Y A349652 Cf. A349561, A349653.
%K A349652 sign
%O A349652 0,3
%A A349652 _Seiichi Manyama_, Nov 23 2021