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

%I A344775 #30 Jun 17 2021 22:14:24
%S A344775 1,1,3,7,23,75,296,1222,5699,28160,151857,867356,5302073,34176364,
%T A344775 232932946,1665341260,12487204067,97743060158,797730561155,
%U A344775 6768022876452,59606300409007,543773719267894,5131560749880622,50012790651415626,502782861641973256,5206962982060933623
%N A344775 a(n) is the number of 2-balanced partitions of a set of n elements.
%C A344775 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.
%C A344775 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).
%H A344775 Alois P. Heinz, <a href="/A344775/b344775.txt">Table of n, a(n) for n = 0..576</a>
%H A344775 Francesca Aicardi, <a href="https://www.researchgate.net/publication/352438417_BALANCED_PARTITIONS">Balanced partitions</a>, preprint on researchgate, 2021.
%F A344775 a(n) = Sum_{k=0..floor(n/2)} A343254(n,k).
%e A344775 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.
%Y A344775 Row sums of A343254.
%Y A344775 Cf. A000110 (Bell numbers).
%K A344775 nonn
%O A344775 0,3
%A A344775 _Francesca Aicardi_, May 28 2021
%E A344775 a(19)-a(25) from _Alois P. Heinz_, Jun 16 2021