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A349655 E.g.f. satisfies: A(x)^3 * log(A(x)) = exp(x) - 1.

%I A349655 #25 Feb 16 2025 08:34:02
%S A349655 1,1,-4,50,-981,26632,-924526,39114343,-1952373450,112321286934,
%T A349655 -7318049389727,532602419776770,-42825957593127770,
%U A349655 3770431528821292441,-360734325565272740984,37267364164988863692782,-4134667018838875759388749,490302545213321842575157140
%N A349655 E.g.f. satisfies: A(x)^3 * log(A(x)) = exp(x) - 1.
%H A349655 Seiichi Manyama, <a href="/A349655/b349655.txt">Table of n, a(n) for n = 0..337</a>
%H A349655 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349655 a(n) = Sum_{k=0..n} (-3*k+1)^(k-1) * Stirling2(n,k).
%F A349655 E.g.f.: A(x) = exp( LambertW(3*(exp(x) - 1))/3 ).
%F A349655 G.f.: Sum_{k>=0} (-3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 - j*x).
%F A349655 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
%p A349655 b:= proc(n, m) option remember; `if`(n=0,
%p A349655      (1-3*m)^(m-1), m*b(n-1, m)+b(n-1, m+1))
%p A349655     end:
%p A349655 a:= n-> b(n, 0):
%p A349655 seq(a(n), n=0..21);  # _Alois P. Heinz_, Jul 29 2022
%t A349655 a[n_] := Sum[(-3*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 27 2021 *)
%o A349655 (PARI) a(n) = sum(k=0, n, (-3*k+1)^(k-1)*stirling(n, k, 2));
%o A349655 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(3*(exp(x)-1))/3)))
%o A349655 (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 A349655 Cf. A349651, A349653, A349657.
%Y A349655 Cf. A008277, A349583, A349654.
%K A349655 sign
%O A349655 0,3
%A A349655 _Seiichi Manyama_, Nov 23 2021