A379706 Number of nonempty labeled antichains of subsets of [n] such that the largest subset is of size 2.

0, 0, 1, 10, 97, 1418, 40005, 2350474, 286192257, 71213783154, 35883905262757, 36419649682704418, 74221659280476132145, 303193505953871645554778, 2480118046704094643352342117, 40601989176407026666590990389338, 1329877330167226219547875498464450945, 87134888326188320631048795061602782878050

Offset: 0

Author

John Tyler Rascoe, Dec 30 2024

Keywords

• nonn
• easy

Examples

			a(2) = 1: {{1,2}}.
a(3) = 10: {{1,2}}, {{1,3}}, {{2,3}}, {{1,2},{3}}, {{1,3},{2}}, {{2,3},{1}}, {{1,2},{1,3}}, {{1,2},{2,3}}, {{1,3},{2,3}}, {{1,2},{1,3},{2,3}}.
		

Crossrefs

Cf. A000225, A000372, A036239, A126883, A379707, (column k=2 of A379712).

Programs

Python

from math import comb
def A379706(n): return sum(comb(n,i)*(2**comb(n-i,2) - 1) for i in range(n-1))

Formula

a(n) = Sum_{i=0..n-2} binomial(n,i) * (2^binomial(n-i,2) - 1).