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A349651 E.g.f. satisfies: A(x)^(A(x)^3) = 1 + x.

%I A349651 #25 Feb 16 2025 08:34:02
%S A349651 1,1,-6,81,-1776,54240,-2125122,101631558,-5739235128,373745355984,
%T A349651 -27572590788480,2272763834553168,-207013811669644680,
%U A349651 20647997125333476912,-2238256520486195804280,262010379635788799196360,-32939968662220720559744448
%N A349651 E.g.f. satisfies: A(x)^(A(x)^3) = 1 + x.
%H A349651 Seiichi Manyama, <a href="/A349651/b349651.txt">Table of n, a(n) for n = 0..331</a>
%H A349651 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349651 a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * |Stirling1(n,k)|.
%F A349651 E.g.f. A(x) = -Sum_{k>=0} (3*k-1)^(k-1) * (-log(1+x))^k / k!.
%F A349651 E.g.f.: A(x) = ( 3*log(1+x)/LambertW(3*log(1+x)) )^(1/3).
%F A349651 a(n) ~ -(-1)^n * n^(n-1) * exp(1/6 - n + n*exp(-1)/3) / (sqrt(3) * (exp(exp(-1)/3) - 1)^(n - 1/2)). - _Vaclav Kotesovec_, Nov 24 2021
%t A349651 nmax = 20; A[_] = 1;
%t A349651 Do[A[x_] = (1 + x)^(1/A[x]^3) + O[x]^(nmax+1) // Normal, {nmax}];
%t A349651 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A349651 (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (3*k-1)^(k-1)*abs(stirling(n, k, 1)));
%o A349651 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(-sum(k=0, N, (3*k-1)^(k-1)*(-log(1+x))^k/k!)))
%o A349651 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((3*log(1+x)/lambertw(3*log(1+x)))^(1/3)))
%Y A349651 Cf. A349653, A349655, A349657.
%Y A349651 Cf. A120980, A349650.
%K A349651 sign
%O A349651 0,3
%A A349651 _Seiichi Manyama_, Nov 23 2021