cp's OEIS Frontend

A379707 Number of nonempty labeled antichains of subsets of [n] such that all subsets except possibly those of the largest size are disjoint.

%I A379707 #20 May 06 2025 11:30:47
%S A379707 1,2,5,19,133,2605,1128365,68731541392,1180735736455875189405,
%T A379707 170141183460507927984536600089529165335,
%U A379707 7237005577335553223087828975127304180898559033209149835788539833222132944557
%N A379707 Number of nonempty labeled antichains of subsets of [n] such that all subsets except possibly those of the largest size are disjoint.
%F A379707 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)).
%e A379707 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.
%o A379707 (Python)
%o A379707 from math import comb
%o A379707 def rS2(n,k,m):
%o A379707     if n < 1 and k < 1: return 1
%o A379707     elif n < 1 or k < 1: return 0
%o A379707     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)
%o A379707 def A229223(n,k):
%o A379707     return sum(rS2(n,x,k) for x in range(n+1))
%o A379707 def A379707(n):
%o A379707     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))
%Y A379707 Cf. A000372, A014466, A229223, A245567, A305844, A379706.
%K A379707 nonn,easy
%O A379707 0,2
%A A379707 _John Tyler Rascoe_, Dec 30 2024