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A343666
Number of partitions of an n-set without blocks of size 6.
7
1, 1, 2, 5, 15, 52, 202, 870, 4084, 20727, 112825, 654546, 4026487, 26145511, 178550986, 1278168860, 9564026947, 74615547996, 605593775899, 5103054929621, 44564754448972, 402677613100491, 3759094788129312, 36205919126040190, 359340174509911325, 3670825700549853053
OFFSET
0,3
FORMULA
E.g.f.: exp(exp(x) - 1 - x^6/6!).
a(n) = n! * Sum_{k=0..floor(n/6)} (-1)^k * Bell(n-6*k) / ((n-6*k)! * k! * (6!)^k).
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, add(
`if`(j=6, 0, a(n-j)*binomial(n-1, j-1)), j=1..n))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Apr 25 2021
MATHEMATICA
nmax = 25; CoefficientList[Series[Exp[Exp[x] - 1 - x^6/6!], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[(-1)^k BellB[n - 6 k]/((n - 6 k)! k! (6!)^k), {k, 0, Floor[n/6]}], {n, 0, 25}]
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}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 25 2021
STATUS
approved