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A288790 Number of blocks of size >= eight in all set partitions of n.

%I A288790 #16 Jun 26 2022 08:59:36
%S A288790 1,10,101,947,8670,79249,730745,6838642,65197797,634656360,6316333291,
%T A288790 64318009411,670336612614,7151290120037,78085166445577,
%U A288790 872478836270306,9972817907218608,116575837400037486,1393037460835481622,17010118386233081680,212160149063581345610
%N A288790 Number of blocks of size >= eight in all set partitions of n.
%H A288790 Alois P. Heinz, <a href="/A288790/b288790.txt">Table of n, a(n) for n = 8..575</a>
%H A288790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A288790 a(n) = Bell(n+1) - Sum_{j=0..7} binomial(n,j) * Bell(n-j).
%F A288790 a(n) = Sum_{j=0..n-8} binomial(n,j) * Bell(j).
%F A288790 E.g.f.: (exp(x) - Sum_{k=0..7} x^k/k!) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 26 2022
%p A288790 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A288790       b(n-j)*binomial(n-1, j-1), j=1..n))
%p A288790     end:
%p A288790 g:= proc(n, k) option remember; `if`(n<k, 0,
%p A288790       g(n, k+1) +binomial(n, k)*b(n-k))
%p A288790     end:
%p A288790 a:= n-> g(n, 8):
%p A288790 seq(a(n), n=8..30);
%t A288790 Table[Sum[Binomial[n, j] BellB[j], {j, 0, n - 8}], {n, 8, 30}] (* _Indranil Ghosh_, Jul 06 2017 *)
%o A288790 (Python)
%o A288790 from sympy import bell, binomial
%o A288790 def a(n): return sum(binomial(n, j)*bell(j) for j in range(n - 7))
%o A288790 print([a(n) for n in range(8, 31)]) # _Indranil Ghosh_, Jul 06 2017
%Y A288790 Column k=8 of A283424.
%Y A288790 Cf. A000110.
%K A288790 nonn
%O A288790 8,2
%A A288790 _Alois P. Heinz_, Jun 15 2017