A286896 - OEIS

%I #13 Jul 23 2021 04:13:16

%S 1,3,17,137,1395,16955,237426,3740609,65197797,1241499241,25577181324,

%T 565688751435,13346516581331,334144326030052,8837737924901855,

%U 245998212661731213,7182425756528424275,219332432679783740235,6987451758608249737342,231704015156531645221237

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

%H Alois P. Heinz, <a href="/A286896/b286896.txt">Table of n, a(n) for n = 0..445</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(n) = Sum_{j=0..n} binomial(2n,j) * Bell(j).

%F a(n) = A283424(2n,n).

%F 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 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 b:= proc(n, k) option remember; `if`(k>n, 0,

%p       binomial(n, k)*combinat[bell](n-k)+b(n, k+1))

%p     end:

%p a:= n-> b(2*n, n):

%p seq(a(n), n=0..25);

%t a[n_] := Sum[Binomial[2 n, j] BellB[j], {j, 0, n}];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 28 2018 *)

%Y Cf. A000110, A283424.

%K nonn

%O 0,2

%A _Alois P. Heinz_, May 15 2017