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%I #5 Oct 08 2023 09:25:41
%S 1,15,395,15525,813671,53306325,4190730335,384368222925,
%T 40289992211591,4751157347330085,622528350091484975,
%U 89724601853904952125,14107579506569655343511,2403010007367884873188245,440801776092151383251034815,86635186648455606881413582125,18162432724968339044562784395431
%N Expansion of e.g.f. 1 / (-4 + Sum_{k=1..5} exp(-k*x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + ... + 5^k) * a(n-k).
%t nmax = 16; CoefficientList[Series[1/(-4 + Sum[Exp[-k x], {k, 1, 5}]), {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 + 5^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 16}]
%Y Cf. A001552, A004703, A005923, A319509, A366298, A366299, A366301, A366302.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Oct 06 2023