A288785 - OEIS

%I #22 Jun 24 2022 17:17:48

%S 1,5,26,137,750,4307,25996,164825,1096217,7633650,55549664,421599778,

%T 3331027887,27349472297,232967157736,2055635993935,18762063976810,

%U 176896220650029,1720762736285790,17249873608817569,178010337967774511,1889129778601708612

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

%H Alois P. Heinz, <a href="/A288785/b288785.txt">Table of n, a(n) for n = 3..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..2} binomial(n,j) * Bell(n-j).

%F a(n) = Sum_{j=0..n-3} binomial(n,j) * Bell(j).

%F a(n) = Sum_{k=1..n} k * A355144(n,k). - _Alois P. Heinz_, Jun 20 2022

%F E.g.f.: (exp(x) - 1 - x - x^2/2) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 24 2022

%e a(4) = 5: 1234, 123|4, 124|3, 134|2, 1|234.

%e 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 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 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, 3):

%p seq(a(n), n=3..30);

%p # second Maple program:

%p b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+[0,

%p      `if`(j>2, p[1], 0)])(b(n-j)*binomial(n-1, j-1)), j=1..n))

%p     end:

%p a:= n-> b(n)[2]:

%p seq(a(n), n=3..30);  # _Alois P. Heinz_, Jan 06 2022

%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, 3];

%t Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, May 28 2018, from Maple *)

%Y Column k=3 of A283424.

%Y Cf. A000110, A355144.

%K nonn

%O 3,2

%A _Alois P. Heinz_, Jun 15 2017