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A343542
Number of ways to partition n labeled elements into sets of different sizes of at least 5.
3
1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 463, 793, 3004, 5006, 14444, 23817, 62323, 14805403, 35175993, 177791475, 745977222, 2333540804, 7589340982, 29027728612, 81515120641, 23232813583331, 69799133324911, 436678552247551, 2215090972333651, 13529994077951557, 48863594588239153
OFFSET
0,12
LINKS
FORMULA
E.g.f.: Product_{k>=5} (1 + x^k/k!).
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i>n, 0, b(n, i+1)+binomial(n, i)*b(n-i, i+1)))
end:
a:= n-> b(n, 5):
seq(a(n), n=0..31); # Alois P. Heinz, Apr 28 2021
MATHEMATICA
nmax = 31; CoefficientList[Series[Product[(1 + x^k/k!), {k, 5, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -(n - 1)! Sum[DivisorSum[k, # (-#!)^(-k/#) &, # > 4 &] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 31}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 28 2021
STATUS
approved