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A286896 Number of blocks of size >= n in all set partitions of [2n].

%I A286896 #13 Jul 23 2021 04:13:16
%S A286896 1,3,17,137,1395,16955,237426,3740609,65197797,1241499241,25577181324,
%T A286896 565688751435,13346516581331,334144326030052,8837737924901855,
%U A286896 245998212661731213,7182425756528424275,219332432679783740235,6987451758608249737342,231704015156531645221237
%N A286896 Number of blocks of size >= n in all set partitions of [2n].
%H A286896 Alois P. Heinz, <a href="/A286896/b286896.txt">Table of n, a(n) for n = 0..445</a>
%H A286896 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A286896 a(n) = Sum_{j=0..n} binomial(2n,j) * Bell(j).
%F A286896 a(n) = A283424(2n,n).
%F A286896 a(n) ~ 2^(2*n) * exp(n/LambertW(n) - n - 1) * n^(n - 1/2) / (sqrt(Pi*(1 + LambertW(n))) * LambertW(n)^n). - _Vaclav Kotesovec_, Jul 23 2021
%e A286896 a(2) = 17: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34. Here three set partitions contain 2 blocks of size 2.
%p A286896 b:= proc(n, k) option remember; `if`(k>n, 0,
%p A286896       binomial(n, k)*combinat[bell](n-k)+b(n, k+1))
%p A286896     end:
%p A286896 a:= n-> b(2*n, n):
%p A286896 seq(a(n), n=0..25);
%t A286896 a[n_] := Sum[Binomial[2 n, j] BellB[j], {j, 0, n}];
%t A286896 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 28 2018 *)
%Y A286896 Cf. A000110, A283424.
%K A286896 nonn
%O A286896 0,2
%A A286896 _Alois P. Heinz_, May 15 2017