A288788 - OEIS

%I #13 Jun 26 2022 08:21:53

%S 1,8,65,502,3851,29921,237426,1932529,16173029,139320277,1235847277,

%T 11288120480,106132359679,1026681599731,10212591089574,

%U 104393925768077,1095895294558168,11806719056706773,130457490607638988,1477428802636263486,17138268233851671782

%N Number of blocks of size >= 6 in all set partitions of n.

%H Alois P. Heinz, <a href="/A288788/b288788.txt">Table of n, a(n) for n = 6..575</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(n) = Bell(n+1) - Sum_{j=0..5} binomial(n,j) * Bell(n-j).

%F a(n) = Sum_{j=0..n-6} binomial(n,j) * Bell(j).

%F E.g.f.: (exp(x) - Sum_{k=0..5} x^k/k!) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 26 2022

%p b:= proc(n) option remember; `if`(n=0, 1, add(

%p       b(n-j)*binomial(n-1, j-1), j=1..n))

%p     end:

%p g:= proc(n, k) option remember; `if`(n<k, 0,

%p       g(n, k+1) +binomial(n, k)*b(n-k))

%p     end:

%p a:= n-> g(n, 6):

%p seq(a(n), n=6..30);

%t b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Binomial[n-1, j-1], {j, 1, n}]];

%t g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k + 1] + Binomial[n, k]*b[n - k]];

%t a[n_] := g[n, 6];

%t Table[a[n], {n, 6, 30}] (* _Jean-François Alcover_, May 28 2018, from Maple *)

%Y Column k=6 of A283424.

%Y Cf. A000110.

%K nonn

%O 6,2

%A _Alois P. Heinz_, Jun 15 2017