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%I #5 Oct 08 2023 09:25:50
%S 1,21,791,44541,3344327,313883661,35351663831,4645129190541,
%T 697553757742247,117844709608925901,22120757207544654071,
%U 4567542244067740041741,1028853921587420129556167,251065459281889114259025741,65978874409961267115296383511,18577448234544937135538443584141
%N Expansion of e.g.f. 1 / (-5 + Sum_{k=1..6} exp(-k*x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + ... + 6^k) * a(n-k).
%t nmax = 15; CoefficientList[Series[1/(-5 + Sum[Exp[-k x], {k, 1, 6}]), {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 + 6^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
%Y Cf. A001553, A004704, A005923, A319509, A366298, A366299, A366300, A366302.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Oct 06 2023