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A288786 Number of blocks of size >= four in all set partitions of n.

%I A288786 #16 Jun 26 2022 05:52:15
%S A288786 1,6,37,225,1395,8944,59585,413117,2981310,22380814,174600298,
%T A288786 1413841252,11868587577,103155618776,927141821215,8606806236367,
%U A288786 82430269073469,813600584094320,8267450613029789,86406853732930699,927993270700444588,10232636504064477996
%N A288786 Number of blocks of size >= four in all set partitions of n.
%H A288786 Alois P. Heinz, <a href="/A288786/b288786.txt">Table of n, a(n) for n = 4..575</a>
%H A288786 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A288786 a(n) = Bell(n+1) - Sum_{j=0..3} binomial(n,j) * Bell(n-j).
%F A288786 a(n) = Sum_{j=0..n-4} binomial(n,j) * Bell(j).
%F A288786 E.g.f.: (exp(x) - Sum_{k=0..3} x^k/k!) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 25 2022
%p A288786 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A288786       b(n-j)*binomial(n-1, j-1), j=1..n))
%p A288786     end:
%p A288786 g:= proc(n, k) option remember; `if`(n<k, 0,
%p A288786       g(n, k+1) +binomial(n, k)*b(n-k))
%p A288786     end:
%p A288786 a:= n-> g(n, 4):
%p A288786 seq(a(n), n=4..30);
%p A288786 # second Maple program:
%p A288786 b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+[0,
%p A288786      `if`(j>3, p[1], 0)])(b(n-j)*binomial(n-1, j-1)), j=1..n))
%p A288786     end:
%p A288786 a:= n-> b(n)[2]:
%p A288786 seq(a(n), n=4..30);  # _Alois P. Heinz_, Jan 06 2022
%t A288786 b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Binomial[n-1, j-1], {j, 1, n}]];
%t A288786 g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k+1] + Binomial[n, k]*b[n - k]];
%t A288786 a[n_] := g[n, 4];
%t A288786 Table[a[n], {n, 4, 30}] (* _Jean-François Alcover_, May 28 2018, from Maple *)
%Y A288786 Column k=4 of A283424.
%Y A288786 Cf. A000110.
%K A288786 nonn
%O A288786 4,2
%A A288786 _Alois P. Heinz_, Jun 15 2017