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E.g.f. satisfies: A(x)^(A(x)^3) = 1 + x.
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%I #24 Mar 04 2024 08:47:31

%S 1,1,-6,81,-1776,54240,-2125122,101631558,-5739235128,373745355984,

%T -27572590788480,2272763834553168,-207013811669644680,

%U 20647997125333476912,-2238256520486195804280,262010379635788799196360,-32939968662220720559744448

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

%H Seiichi Manyama, <a href="/A349651/b349651.txt">Table of n, a(n) for n = 0..331</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * |Stirling1(n,k)|.

%F E.g.f. A(x) = -Sum_{k>=0} (3*k-1)^(k-1) * (-log(1+x))^k / k!.

%F E.g.f.: A(x) = ( 3*log(1+x)/LambertW(3*log(1+x)) )^(1/3).

%F 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 nmax = 20; A[_] = 1;

%t Do[A[x_] = (1 + x)^(1/A[x]^3) + O[x]^(nmax+1) // Normal, {nmax}];

%t CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)

%o (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (3*k-1)^(k-1)*abs(stirling(n, k, 1)));

%o (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 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace((3*log(1+x)/lambertw(3*log(1+x)))^(1/3)))

%Y Cf. A349653, A349655, A349657.

%Y Cf. A120980, A349650.

%K sign

%O 0,3

%A _Seiichi Manyama_, Nov 23 2021