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A288788 Number of blocks of size >= 6 in all set partitions of n.

%I A288788 #13 Jun 26 2022 08:21:53
%S A288788 1,8,65,502,3851,29921,237426,1932529,16173029,139320277,1235847277,
%T A288788 11288120480,106132359679,1026681599731,10212591089574,
%U A288788 104393925768077,1095895294558168,11806719056706773,130457490607638988,1477428802636263486,17138268233851671782
%N A288788 Number of blocks of size >= 6 in all set partitions of n.
%H A288788 Alois P. Heinz, <a href="/A288788/b288788.txt">Table of n, a(n) for n = 6..575</a>
%H A288788 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A288788 a(n) = Bell(n+1) - Sum_{j=0..5} binomial(n,j) * Bell(n-j).
%F A288788 a(n) = Sum_{j=0..n-6} binomial(n,j) * Bell(j).
%F A288788 E.g.f.: (exp(x) - Sum_{k=0..5} x^k/k!) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 26 2022
%p A288788 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A288788       b(n-j)*binomial(n-1, j-1), j=1..n))
%p A288788     end:
%p A288788 g:= proc(n, k) option remember; `if`(n<k, 0,
%p A288788       g(n, k+1) +binomial(n, k)*b(n-k))
%p A288788     end:
%p A288788 a:= n-> g(n, 6):
%p A288788 seq(a(n), n=6..30);
%t A288788 b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Binomial[n-1, j-1], {j, 1, n}]];
%t A288788 g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k + 1] + Binomial[n, k]*b[n - k]];
%t A288788 a[n_] := g[n, 6];
%t A288788 Table[a[n], {n, 6, 30}] (* _Jean-François Alcover_, May 28 2018, from Maple *)
%Y A288788 Column k=6 of A283424.
%Y A288788 Cf. A000110.
%K A288788 nonn
%O A288788 6,2
%A A288788 _Alois P. Heinz_, Jun 15 2017