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

Author

Francesca Aicardi, May 28 2021

Keywords

• nonn

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).

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Francesca Aicardi, Balanced partitions, preprint on researchgate, 2021.

Crossrefs

Row sums of A343254.

Cf. A000110 (Bell numbers).

Formula

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

Extensions

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