cp's OEIS Frontend

A288789 Number of blocks of size >= 7 in all set partitions of n.

%I A288789 #15 Jun 26 2022 07:32:40
%S A288789 1,9,82,701,5897,49854,427597,3740609,33479542,307119477,2890138160,
%T A288789 27911144971,276632735047,2813333368854,29349063282197,
%U A288789 313940448544057,3441759044602385,38652680805862224,444450158120668786,5229815283321976222,62942722623990478840
%N A288789 Number of blocks of size >= 7 in all set partitions of n.
%H A288789 Alois P. Heinz, <a href="/A288789/b288789.txt">Table of n, a(n) for n = 7..575</a>
%H A288789 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A288789 a(n) = Bell(n+1) - Sum_{j=0..6} binomial(n,j) * Bell(n-j).
%F A288789 a(n) = Sum_{j=0..n-7} binomial(n,j) * Bell(j).
%F A288789 E.g.f.: (exp(x) - Sum_{k=0..6} x^k/k!) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 26 2022
%p A288789 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A288789       b(n-j)*binomial(n-1, j-1), j=1..n))
%p A288789     end:
%p A288789 g:= proc(n, k) option remember; `if`(n<k, 0,
%p A288789       g(n, k+1) +binomial(n, k)*b(n-k))
%p A288789     end:
%p A288789 a:= n-> g(n, 7):
%p A288789 seq(a(n), n=7..30);
%p A288789 # second Maple program:
%p A288789 b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+
%p A288789       `if`(j>6, [0, p[1]], 0))(b(n-j)*binomial(n-1, j-1)), j=1..n))
%p A288789     end:
%p A288789 a:= n-> b(n)[2]:
%p A288789 seq(a(n), n=7..30);  # _Alois P. Heinz_, Jun 26 2022
%t A288789 b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j]*Binomial[n-1, j-1], {j, 1, n}]];
%t A288789 g[n_, k_] := g[n, k] = If[n < k, 0, g[n, k + 1] + Binomial[n, k]*b[n - k]];
%t A288789 a[n_] := g[n, 7];
%t A288789 Table[a[n], {n, 7, 30}] (* _Jean-François Alcover_, May 28 2018, from Maple *)
%Y A288789 Column k=7 of A283424.
%Y A288789 Cf. A000110.
%K A288789 nonn
%O A288789 7,2
%A A288789 _Alois P. Heinz_, Jun 15 2017