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%I #17 Nov 18 2023 08:26:47
%S 1,1,7,73,1063,20041,464167,12752713,405439783,14641740361,
%T 592050220327,26499885031753,1300723181304103,69470729022993481,
%U 4010891467932629287,248920020505516389193,16525139232054244298023,1168557027163488299171401
%N Expansion of e.g.f. 1/(2 - exp(3*x))^(1/3).
%F a(n) = Sum_{k=0..n} 3^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling2(n,k).
%F a(n) ~ n! * 3^n / (2^(1/3) * Gamma(1/3) * n^(2/3) * log(2)^(n + 1/3)). - _Vaclav Kotesovec_, Mar 05 2022
%F From _Seiichi Manyama_, Nov 18 2023: (Start)
%F a(0) = 1; a(n) = Sum_{k=1..n} 3^k * (1 - 2/3 * k/n) * binomial(n,k) * a(n-k).
%F a(0) = 1; a(n) = a(n-1) - 2*Sum_{k=1..n-1} (-3)^k * binomial(n-1,k) * a(n-k). (End)
%t m = 17; Range[0, m]! * CoefficientList[Series[(2 - Exp[3*x])^(-1/3), {x, 0, m}], x] (* _Amiram Eldar_, Mar 05 2022 *)
%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(3*x))^(1/3)))
%o (PARI) a(n) = sum(k=0, n, 3^(n-k)*prod(j=0, k-1, 3*j+1)*stirling(n, k, 2));
%Y Cf. A000670, A352117, A352119.
%Y Cf. A352070.
%K nonn
%O 0,3
%A _Seiichi Manyama_, Mar 05 2022
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