A344775 a(n) is the number of 2-balanced partitions of a set of n elements.

1, 1, 3, 7, 23, 75, 296, 1222, 5699, 28160, 151857, 867356, 5302073, 34176364, 232932946, 1665341260, 12487204067, 97743060158, 797730561155, 6768022876452, 59606300409007, 543773719267894, 5131560749880622, 50012790651415626, 502782861641973256, 5206962982060933623

Offset

0,3

Comments

A 2-balanced partition is a partition of a set which is the union of three subsets, with the property that the cardinality of the first two subsets are equal (possibly zero), and each block contains the same number (possibly zero) of elements from the first and from the second subset.

a(n) is calculated as the sum of the numbers b(n,k) (A343254) of 2-balanced partitions of a set of n elements in which the first and the second subsets have cardinality k. The sum runs over all integers k from zero to floor(n/2).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..576

Francesca Aicardi, Balanced partitions, preprint on researchgate, 2021.

Formula

a(n) = Sum_{k=0..floor(n/2)} A343254(n,k).

Example

For n=3, a(3) = b(3,0) + b(3,1). b(3,0) is the number of partitions of a set of three elements (all elements lie in the third subset), i.e., b(3,0) = Bell(3) = 5. b(3,1) is the number of 2-balanced partitions of a set {p,q,r} in which the first and the second subsets, say {p} and {q}, have cardinality 1. There are only two 2-balanced partitions: {{p,q},{r}}, and {{p,q,r}}. So, b(3,1)=2 and a(3)=7.

Crossrefs

Row sums of A343254.

Cf. A000110 (Bell numbers).

Sequence in context: A148709 A148710 A148711 * A205481 A341071 A148712

Adjacent sequences: A344772 A344773 A344774 * A344776 A344777 A344778

Keyword

nonn

Author

Francesca Aicardi, May 28 2021

Extensions

a(19)-a(25) from Alois P. Heinz, Jun 16 2021

Status

approved