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

1, 3, 17, 137, 1395, 16955, 237426, 3740609, 65197797, 1241499241, 25577181324, 565688751435, 13346516581331, 334144326030052, 8837737924901855, 245998212661731213, 7182425756528424275, 219332432679783740235, 6987451758608249737342, 231704015156531645221237

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..445

Wikipedia, Partition of a set

Formula

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

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

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

Example

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.

Maple

b:= proc(n, k) option remember; `if`(k>n, 0,
      binomial(n, k)*combinat[bell](n-k)+b(n, k+1))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=0..25);

Mathematica

a[n_] := Sum[Binomial[2 n, j] BellB[j], {j, 0, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 28 2018 *)

Crossrefs

Cf. A000110, A283424.

Sequence in context: A350736 A231909 A331688 * A244432 A219503 A230387

Adjacent sequences: A286893 A286894 A286895 * A286897 A286898 A286899

Keyword

nonn

Author

Alois P. Heinz, May 15 2017

Status

approved