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A288792 Number of blocks of size >= ten in all set partitions of n.

%I A288792 #16 Jun 26 2022 09:01:39
%S A288792 1,12,145,1600,17032,179132,1883117,19929390,213332101,2316793121,
%T A288792 25577181324,287421068697,3290394397097,38393883291996,
%U A288792 456753452800691,5540597439008861,68530489547341697,864218608315007230,11109867095322262250,145563654356205885737
%N A288792 Number of blocks of size >= ten in all set partitions of n.
%H A288792 Alois P. Heinz, <a href="/A288792/b288792.txt">Table of n, a(n) for n = 10..576</a>
%H A288792 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A288792 a(n) = Bell(n+1) - Sum_{j=0..9} binomial(n,j) * Bell(n-j).
%F A288792 a(n) = Sum_{j=0..n-10} binomial(n,j) * Bell(j).
%F A288792 E.g.f.: (exp(x) - Sum_{k=0..9} x^k/k!) * exp(exp(x) - 1). - _Ilya Gutkovskiy_, Jun 26 2022
%p A288792 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A288792       b(n-j)*binomial(n-1, j-1), j=1..n))
%p A288792     end:
%p A288792 g:= proc(n, k) option remember; `if`(n<k, 0,
%p A288792       g(n, k+1) +binomial(n, k)*b(n-k))
%p A288792     end:
%p A288792 a:= n-> g(n, 10):
%p A288792 seq(a(n), n=10..30);
%t A288792 Table[Sum[Binomial[n, j] BellB[j], {j, 0, n - 10}], {n, 10, 30}] (* _Indranil Ghosh_, Jul 06 2017 *)
%o A288792 (Python)
%o A288792 from sympy import bell, binomial
%o A288792 def a(n): return sum([binomial(n, j)*bell(j) for j in range(n - 9)])
%o A288792 print([a(n) for n in range(10, 31)]) # _Indranil Ghosh_, Jul 06 2017
%Y A288792 Column k=10 of A283424.
%Y A288792 Cf. A000110.
%K A288792 nonn
%O A288792 10,2
%A A288792 _Alois P. Heinz_, Jun 15 2017