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A349650 E.g.f. satisfies: A(x)^(A(x)^2) = 1 + x.

%I A349650 #23 Feb 16 2025 08:34:02
%S A349650 1,1,-4,36,-532,11040,-295188,9655772,-373422320,16666348464,
%T A349650 -843095987520,47669276120928,-2979044176833888,203906085094788960,
%U A349650 -15170476121142482112,1218972837861962011200,-105202043767190506428672,9705514148732971389369600
%N A349650 E.g.f. satisfies: A(x)^(A(x)^2) = 1 + x.
%H A349650 Seiichi Manyama, <a href="/A349650/b349650.txt">Table of n, a(n) for n = 0..347</a>
%H A349650 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349650 a(n) = (-1)^(n-1) * Sum_{k=0..n} (2*k-1)^(k-1) * |Stirling1(n,k)|.
%F A349650 E.g.f. A(x) = -Sum_{k>=0} (2*k-1)^(k-1) * (-log(1+x))^k / k!.
%F A349650 E.g.f.: A(x) = ( 2*log(1+x)/LambertW(2*log(1+x)) )^(1/2).
%F A349650 a(n) ~ -(-1)^n * n^(n-1) * exp(n*(exp(-1)/2 - 1)) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n - 1/2)). - _Vaclav Kotesovec_, Nov 24 2021
%t A349650 nmax = 20; A[_] = 1;
%t A349650 Do[A[x_] = (1 + x)^(1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax}];
%t A349650 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A349650 (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (2*k-1)^(k-1)*abs(stirling(n, k, 1)));
%o A349650 (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 A349650 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((2*log(1+x)/lambertw(2*log(1+x)))^(1/2)))
%Y A349650 Cf. A349652, A349654, A349656.
%Y A349650 Cf. A120980, A349651.
%K A349650 sign
%O A349650 0,3
%A A349650 _Seiichi Manyama_, Nov 23 2021