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A349657 E.g.f. satisfies: A(x)^3 * log(A(x)) = 1 - exp(-x).

%I A349657 #23 Feb 16 2025 08:34:02
%S A349657 1,1,-6,80,-1751,53402,-2088528,99680667,-5617170700,365003288652,
%T A349657 -26868393676609,2209797209486528,-200828403704351068,
%U A349657 19986049281174575497,-2161617056877509895386,252467067400866652634004,-31668302130310076212791823
%N A349657 E.g.f. satisfies: A(x)^3 * log(A(x)) = 1 - exp(-x).
%H A349657 Seiichi Manyama, <a href="/A349657/b349657.txt">Table of n, a(n) for n = 0..331</a>
%H A349657 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349657 a(n) = (-1)^(n-1) * Sum_{k=0..n} (3*k-1)^(k-1) * Stirling2(n,k).
%F A349657 E.g.f.: A(x) = exp( LambertW(3*(1 - exp(-x)))/3 ).
%F A349657 G.f.: Sum_{k>=0} (-3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
%F A349657 a(n) ~ -(-1)^n * sqrt(3*exp(1) + 1) * sqrt(-log(3) + log(3 + exp(-1))) * n^(n-1) / (3 * exp(n + 1/3) * (-log(3) + log(3*exp(1) + 1) - 1)^n). - _Vaclav Kotesovec_, Nov 24 2021
%t A349657 nmax = 20; A[_] = 1;
%t A349657 Do[A[x_] = Exp[(Exp[x]-1)/(Exp[x]*A[x]^3)]+O[x]^(nmax+1)//Normal, {nmax}];
%t A349657 CoefficientList[A[x], x]*Range[0, nmax]! (* _Jean-François Alcover_, Mar 04 2024 *)
%o A349657 (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (3*k-1)^(k-1)*stirling(n, k, 2));
%o A349657 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*(1-exp(-x)))/3)))
%o A349657 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-3*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))
%Y A349657 Cf. A349651, A349653, A349655.
%Y A349657 Cf. A349585, A349656.
%K A349657 sign
%O A349657 0,3
%A A349657 _Seiichi Manyama_, Nov 23 2021