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%I #5 Oct 08 2023 09:25:21
%S 1,6,58,828,15766,375276,10719118,357202068,13603819126,582854637276,
%T 27747071520478,1453003753611108,83005119616449286,
%U 5136947527401250476,342365553703113120238,24447711909762202272948,1862151878019906517540246,150702660087903415402794876,12913688931657425188926182398
%N Expansion of e.g.f. 1 / (-2 + Sum_{k=1..3} exp(-k*x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + 3^k) * a(n-k).
%t nmax = 18; CoefficientList[Series[1/(-2 + Sum[Exp[-k x], {k, 1, 3}]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = Sum[(-1)^(k + 1) Binomial[n, k] (1 + 2^k + 3^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%Y Cf. A001550, A004701, A005923, A319509, A366299, A366300, A366301, A366302.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Oct 06 2023