A379707 Number of nonempty labeled antichains of subsets of [n] such that all subsets except possibly those of the largest size are disjoint.
1, 2, 5, 19, 133, 2605, 1128365, 68731541392, 1180735736455875189405, 170141183460507927984536600089529165335, 7237005577335553223087828975127304180898559033209149835788539833222132944557
Offset: 0
Examples
For n < 4 all nonempty labeled antichains are counted. When n=6 antichains such as {{1,2,6},{1,4},{1,5}} are not counted, while {{1,2,4},{1,2,6},{3},{5}} is counted.
Programs
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Python
from math import comb def rS2(n,k,m): if n < 1 and k < 1: return 1 elif n < 1 or k < 1: return 0 else: return k*rS2(n-1,k,m) + rS2(n-1,k-1,m)- comb(n-1,m)*rS2(n-1-m,k-1,m) def A229223(n,k): return sum(rS2(n,x,k) for x in range(n+1)) def A379707(n): return 1+sum(sum(comb(n,i)*(2**comb(n-i,s)-1)*A229223(i,s-1) for i in range(n-s+1)) for s in range(1,n+1))
Formula
a(n) = 1 + Sum_{s=1..n} (Sum_{i=0..n-s} binomial(n,i) * (2^binomial(n-i,s) - 1) * A229223(i,s-1)).