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%I #9 Apr 29 2021 19:43:13
%S 1,1,3,13,75,541,4683,47293,545834,7087243,102247203,1622625313,
%T 28091415135,526854986737,10641264928479,230281282588513,
%U 5315605563021465,130369438065006551,3385496924633886429,92800464391224494215,2677652842774247060805,81123688691904430522831
%N Number of ordered partitions of an n-set without blocks of size 8.
%F E.g.f.: 1 / (2 + x^8/8! - exp(x)).
%p a:= proc(n) option remember; `if`(n=0, 1, add(
%p `if`(j=8, 0, a(n-j)*binomial(n, j)), j=1..n))
%p end:
%p seq(a(n), n=0..21); # _Alois P. Heinz_, Apr 29 2021
%t nmax = 21; CoefficientList[Series[1/(2 + x^8/8! - Exp[x]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 8, 0, Binomial[n, k] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 21}]
%Y Cf. A000670, A032032, A337058, A337059, A343668, A343787, A343788, A343789, A343790, A343792, A343793.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 29 2021
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