A343542 Number of ways to partition n labeled elements into sets of different sizes of at least 5.

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

Author

Ilya Gutkovskiy, Apr 28 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..700

Crossrefs

Cf. A007837, A025150, A032311, A057814, A245790, A341283, A343319.

Programs

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}]

Formula

E.g.f.: Product_{k>=5} (1 + x^k/k!).