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%I #15 Feb 20 2022 06:42:21
%S 1,1,8,69,780,11145,191178,3823785,87406056,2247785073,64228084110,
%T 2018771719569,69221032558956,2571290056399545,102860527370221026,
%U 4408690840306136505,201557641172689004112,9790792086366911655009,503570143277542340304534
%N E.g.f.: 1 / (1 - x * exp(3*x)).
%H Seiichi Manyama, <a href="/A336951/b336951.txt">Table of n, a(n) for n = 0..383</a>
%F a(n) = n! * Sum_{k=0..n} (3 * (n-k))^k / k!.
%F a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * k * 3^(k-1) * a(n-k).
%F a(n) ~ n! * (3/LambertW(3))^n / (1 + LambertW(3)). - _Vaclav Kotesovec_, Aug 09 2021
%t nmax = 18; CoefficientList[Series[1/(1 - x Exp[3 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t Join[{1}, Table[n! Sum[(3 (n - k))^k/k!, {k, 0, n}], {n, 1, 18}]]
%t a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] k 3^(k - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%o (PARI) seq(n)={ Vec(serlaplace(1 / (1 - x*exp(3*x + O(x^n))))) } \\ _Andrew Howroyd_, Aug 08 2020
%Y Column k=3 of A351790.
%Y Cf. A006153, A027471, A235328, A255927, A328182, A336948, A336950, A336952.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Aug 08 2020