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A288791 Number of blocks of size >= nine in all set partitions of n.

%I A288791 #16 Jun 26 2022 09:01:05
%S A288791 1,11,122,1245,12325,121136,1195147,11915997,120572790,1241499241,
%T A288791 13030331671,139549798524,1525923634907,17041290249637,
%U A288791 194394900237176,2264977282222371,26951265841776186,327445918493429897,4060993235341162405,51396034231430455550
%N A288791 Number of blocks of size >= nine in all set partitions of n.
%H A288791 Alois P. Heinz, <a href="/A288791/b288791.txt">Table of n, a(n) for n = 9..575</a>
%H A288791 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A288791 a(n) = Bell(n+1) - Sum_{j=0..8} binomial(n,j) * Bell(n-j).
%F A288791 a(n) = Sum_{j=0..n-9} binomial(n,j) * Bell(j).
%F A288791 E.g.f.: (exp(x) - Sum_{k=0..8} x^k/k!) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 26 2022
%p A288791 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A288791       b(n-j)*binomial(n-1, j-1), j=1..n))
%p A288791     end:
%p A288791 g:= proc(n, k) option remember; `if`(n<k, 0,
%p A288791       g(n, k+1) +binomial(n, k)*b(n-k))
%p A288791     end:
%p A288791 a:= n-> g(n, 9):
%p A288791 seq(a(n), n=9..30);
%t A288791 Table[Sum[Binomial[n, j] BellB[j], {j, 0, n - 9}], {n, 9, 30}] (* _Indranil Ghosh_, Jul 06 2017 *)
%o A288791 (Python)
%o A288791 from sympy import bell, binomial
%o A288791 def a(n): return sum([binomial(n, j)*bell(j) for j in range(n - 8)])
%o A288791 print([a(n) for n in range(9, 31)]) # _Indranil Ghosh_, Jul 06 2017
%Y A288791 Column k=9 of A283424.
%Y A288791 Cf. A000110.
%K A288791 nonn
%O A288791 9,2
%A A288791 _Alois P. Heinz_, Jun 15 2017