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%I #5 Oct 08 2023 09:25:32
%S 1,10,170,4300,145046,6115900,309453710,18267444100,1232400398966,
%T 93535914320620,7887919177776350,731710341934820500,
%U 74046493229735962886,8117679564133907097340,958393800813241073719790,121232569802975799394430500,16357741845227058108680934806,2345072789674603792983906178060
%N Expansion of e.g.f. 1 / (-3 + Sum_{k=1..4} exp(-k*x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + 3^k + 4^k) * a(n-k).
%t nmax = 17; CoefficientList[Series[1/(-3 + Sum[Exp[-k x], {k, 1, 4}]), {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 + 4^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
%Y Cf. A001551, A004702, A005923, A319509, A366298, A366300, A366301, A366302.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Oct 06 2023