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%I #5 Oct 08 2023 09:26:00
%S 1,28,1428,108976,11088924,1410452848,215282610348,38335940184976,
%T 7801807561068444,1786227911508713008,454397569178386774668,
%U 127153351764004535348176,38815768300684586111354364,12836619471891836987050169968,4571701128215207034965181098988,1744488930796462320024115801858576
%N Expansion of e.g.f. 1 / (-6 + Sum_{k=1..7} exp(-k*x)).
%F a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(n,k) * (1 + 2^k + ... + 7^k) * a(n-k).
%t nmax = 15; CoefficientList[Series[1/(-6 + Sum[Exp[-k x], {k, 1, 7}]), {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 + 7^k) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
%Y Cf. A001554, A004705, A005923, A319509, A366298, A366299, A366300, A366301.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Oct 06 2023