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A288785 Number of blocks of size >= three in all set partitions of n.

%I A288785 #22 Jun 24 2022 17:17:48
%S A288785 1,5,26,137,750,4307,25996,164825,1096217,7633650,55549664,421599778,
%T A288785 3331027887,27349472297,232967157736,2055635993935,18762063976810,
%U A288785 176896220650029,1720762736285790,17249873608817569,178010337967774511,1889129778601708612
%N A288785 Number of blocks of size >= three in all set partitions of n.
%H A288785 Alois P. Heinz, <a href="/A288785/b288785.txt">Table of n, a(n) for n = 3..575</a>
%H A288785 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A288785 a(n) = Bell(n+1) - Sum_{j=0..2} binomial(n,j) * Bell(n-j).
%F A288785 a(n) = Sum_{j=0..n-3} binomial(n,j) * Bell(j).
%F A288785 a(n) = Sum_{k=1..n} k * A355144(n,k). - _Alois P. Heinz_, Jun 20 2022
%F A288785 E.g.f.: (exp(x) - 1 - x - x^2/2) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 24 2022
%e A288785 a(4) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
%e A288785 a(5) = 26: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 125|3|4, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 135|2|4, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
%e A288785 a(6) = 137: 123456, 12345|6, 12346|5, ..., 123|456, 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234, ..., 1|256|3|4, 1|2|356|4, 1|2|3|456.
%p A288785 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A288785       b(n-j)*binomial(n-1, j-1), j=1..n))
%p A288785     end:
%p A288785 g:= proc(n, k) option remember; `if`(n<k, 0,
%p A288785       g(n, k+1) +binomial(n, k)*b(n-k))
%p A288785     end:
%p A288785 a:= n-> g(n, 3):
%p A288785 seq(a(n), n=3..30);
%p A288785 # second Maple program:
%p A288785 b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+[0,
%p A288785      `if`(j>2, p[1], 0)])(b(n-j)*binomial(n-1, j-1)), j=1..n))
%p A288785     end:
%p A288785 a:= n-> b(n)[2]:
%p A288785 seq(a(n), n=3..30);  # _Alois P. Heinz_, Jan 06 2022
%t A288785 b[n_] := b[n] = If[n == 0, 1, Sum[b[n-j]*Binomial[n-1, j-1], {j, 1, n}]];
%t A288785 g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k+1] + Binomial[n, k]*b[n - k]];
%t A288785 a[n_] := g[n, 3];
%t A288785 Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, May 28 2018, from Maple *)
%Y A288785 Column k=3 of A283424.
%Y A288785 Cf. A000110, A355144.
%K A288785 nonn
%O A288785 3,2
%A A288785 _Alois P. Heinz_, Jun 15 2017