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%I #7 Apr 25 2021 20:07:47
%S 1,1,2,5,15,52,202,870,4084,20727,112825,654546,4026487,26145511,
%T 178550986,1278168860,9564026947,74615547996,605593775899,
%U 5103054929621,44564754448972,402677613100491,3759094788129312,36205919126040190,359340174509911325,3670825700549853053
%N Number of partitions of an n-set without blocks of size 6.
%F E.g.f.: exp(exp(x) - 1 - x^6/6!).
%F a(n) = n! * Sum_{k=0..floor(n/6)} (-1)^k * Bell(n-6*k) / ((n-6*k)! * k! * (6!)^k).
%p a:= proc(n) option remember; `if`(n=0, 1, add(
%p `if`(j=6, 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^6/6!], {x, 0, nmax}], x] Range[0, nmax]!
%t Table[n! Sum[(-1)^k BellB[n - 6 k]/((n - 6 k)! k! (6!)^k), {k, 0, Floor[n/6]}], {n, 0, 25}]
%t a[n_] := a[n] = If[n == 0, 1, Sum[If[k == 6, 0, Binomial[n - 1, k - 1] a[n - k]], {k, 1, n}]]; Table[a[n], {n, 0, 25}]
%Y Cf. A000110, A000296, A027340, A097514, A124504, A343664, A343665, A343667, A343668, A343669.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Apr 25 2021
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