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%I #7 Apr 25 2021 20:08:00
%S 1,1,2,5,15,52,203,877,4139,21138,115885,677745,4206172,27577513,
%T 190289713,1377315050,10426866782,82350895629,677003941219,
%U 5781485704892,51193839084907,469251258854001,4445769329586348,43475305461354931,438270620701587657,4549243731200717053
%N Number of partitions of an n-set without blocks of size 8.
%F E.g.f.: exp(exp(x) - 1 - x^8/8!).
%F a(n) = n! * Sum_{k=0..floor(n/8)} (-1)^k * Bell(n-8*k) / ((n-8*k)! * k! * (8!)^k).
%p a:= proc(n) option remember; `if`(n=0, 1, add(
%p `if`(j=8, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
%p end:
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Apr 25 2021
%t nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^8/8!], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[n! Sum[(-1)^k BellB[n - 8 k]/((n - 8 k)! k! (8!)^k), {k, 0, Floor[n/8]}], {n, 0, 25}]
%t a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 8, 0, Binomial[n - 1, k - 1] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
%Y Cf. A000110, A000296, A027342, A097514, A124504, A343664, A343665, A343666, A343667, A343669.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 25 2021
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