A389807 a(n) = A389805(2*n).

1, 1, 1, 2, 5, 19, 89, 489, 3122, 22777, 189016, 1761266, 18557581, 218008848, 2850048230, 41383639806, 659845914516, 11512674828157, 218931359938886, 4511589998959414

Offset

1,4

Comments

Number of the complete partitions of T(n) = {-2*n+1, -2*n+3, ..., -3, -1, 1, 3, ..., 2*n-3, 2*n-1}.

A zero-sum integer set is called reducible if it contains a smaller zero-sum subset, otherwise, it is called irreducible.

Partitions of T(n) into a set of irreducible zero-sum subsets are called complete.

Links

Table of n, a(n) for n=1..20.

Hu Junhua, Balanced samples of the initial segment of natural numbers, Chinese Math R & D BBS, Apr 21 2010.

Example

For n = 4, a(4) = 2, since {-7, -5, -3, -1, 1, 3, 5, 7} has 2 complete partitions: {{-1, 1}, {-3, 3}, {-5, 5}, {-7, 7}}, {{-1, 3, 5, -7}, {1, -3, -5, 7}}.
For n = 5, a(5) = 5, since {-9, -7, -5, -3, -1, 1, 3, 5, 7, 9} has 5 complete partitions: {{-1, 1}, {-3, 3}, {-5, 5}, {-7, 7}, {-9, 9}}, {{-1, 1}, {-3, 5, 7, -9}, {3, -5, -7, 9}}, {{-5, 5}, {-1, 3, 7, -9}, {1, -3, -7, 9}}, {{-7, 7}, {1, 3, 5, -9}, {-1, -3, -5, 9}}, {{-9, 9}, {-1, 3, 5, -7}, {1, -3, -5, 7}}.

Crossrefs

Cf. A389802, A389804, A389805, A389806.

Sequence in context: A316700 A124348 A323389 * A268605 A205804 A052324

Adjacent sequences: A389804 A389805 A389806 * A389808 A389809 A389810

Keyword

nonn,more

Author

Hu Junhua, Oct 22 2025

Extensions

a(10)-a(14) from Sean A. Irvine, Oct 21 2025

a(15)-a(20) added (using A389805) by Hu Junhua, Jan 29 2026

Status

approved