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%I #52 Aug 02 2024 13:08:44
%S 1,1,7,109,2665,88981,3768391,193406977,11663021329,808092594505,
%T 63252127883431,5519514702282901,531266903931402937,
%U 55912682968563924829,6387276499619184590695,787104141893585220839401,104074098535487279656795681,14697203663694095986066104337
%N Expansion of e.g.f. exp( (1 - (1 - 9*x)^(1/3))/3 ).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F a(n) = Sum_{k=0..n} (-9)^(n-k) * Stirling1(n,k) * A317996(k) = (-9)^n * Sum_{k=0..n} (1/3)^k * Stirling1(n,k) * Bell_k(-1/3), where Bell_n(x) is n-th Bell polynomial.
%F From _Vaclav Kotesovec_, Aug 02 2024: (Start)
%F a(n) = 18*(n-2)*a(n-1) - 9*(3*n-8)*(3*n-7)*a(n-2) + a(n-3).
%F a(n) ~ Gamma(1/3) * 3^(2*n - 3/2) * n^(n - 5/6) / (sqrt(2*Pi) * exp(n - 1/3)) * (1 - 2*Pi/(3^(3/2)*Gamma(1/3)^2*n^(1/3))). (End)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1-(1-9*x)^(1/3))/3)))
%Y Cf. A080893, A252284, A317996.
%Y Cf. A375174, A375176.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Aug 02 2024