cp's OEIS Frontend

A288787 Number of blocks of size >= five in all set partitions of n.

%I A288787 #16 Jun 26 2022 08:21:19
%S A288787 1,7,50,345,2392,16955,123707,932010,7260709,58509323,487593202,
%T A288787 4199841037,37361858716,342989895895,3246458915947,31653980371254,
%U A288787 317654338317380,3278058775976704,34757921507150964,378372365291381716,4225533329681577846,48375204740642752562
%N A288787 Number of blocks of size >= five in all set partitions of n.
%H A288787 Alois P. Heinz, <a href="/A288787/b288787.txt">Table of n, a(n) for n = 5..575</a>
%H A288787 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A288787 a(n) = Bell(n+1) - Sum_{j=0..4} binomial(n,j) * Bell(n-j).
%F A288787 a(n) = Sum_{j=0..n-5} binomial(n,j) * Bell(j).
%F A288787 E.g.f.: (exp(x) - Sum_{k=0..4} x^k/k!) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 26 2022
%p A288787 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A288787       b(n-j)*binomial(n-1, j-1), j=1..n))
%p A288787     end:
%p A288787 g:= proc(n, k) option remember; `if`(n<k, 0,
%p A288787       g(n, k+1) +binomial(n, k)*b(n-k))
%p A288787     end:
%p A288787 a:= n-> g(n, 5):
%p A288787 seq(a(n), n=5..30);
%t A288787 b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Binomial[n-1, j-1], {j, 1, n}]];
%t A288787 g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k + 1] + Binomial[n, k]*b[n - k]];
%t A288787 a[n_] := g[n, 5];
%t A288787 Table[a[n], {n, 5, 30}] (* _Jean-François Alcover_, May 28 2018, from Maple *)
%Y A288787 Column k=5 of A283424.
%Y A288787 Cf. A000110.
%K A288787 nonn
%O A288787 5,2
%A A288787 _Alois P. Heinz_, Jun 15 2017