OFFSET
5,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 5..575
Wikipedia, Partition of a set
FORMULA
a(n) = Bell(n+1) - Sum_{j=0..4} binomial(n,j) * Bell(n-j).
a(n) = Sum_{j=0..n-5} binomial(n,j) * Bell(j).
E.g.f.: (exp(x) - Sum_{k=0..4} x^k/k!) * exp(exp(x) - 1). - Ilya Gutkovskiy, Jun 26 2022
MAPLE
b:= proc(n) option remember; `if`(n=0, 1, add(
b(n-j)*binomial(n-1, j-1), j=1..n))
end:
g:= proc(n, k) option remember; `if`(n<k, 0,
g(n, k+1) +binomial(n, k)*b(n-k))
end:
a:= n-> g(n, 5):
seq(a(n), n=5..30);
MATHEMATICA
b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Binomial[n-1, j-1], {j, 1, n}]];
g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k + 1] + Binomial[n, k]*b[n - k]];
a[n_] := g[n, 5];
Table[a[n], {n, 5, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 15 2017
STATUS
approved