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A178165 Number of unordered collections of distinct nonempty subsets of an n-element set where each element appears in at most 2 subsets.

%I A178165 #33 Jul 06 2025 03:03:55
%S A178165 1,2,8,59,652,9736,186478,4421018,126317785,4260664251,166884941780,
%T A178165 7489637988545,380861594219460,21739310882945458,1381634777325000263,
%U A178165 97089956842985393297,7497783115765911443879,632884743974716421132084
%N A178165 Number of unordered collections of distinct nonempty subsets of an n-element set where each element appears in at most 2 subsets.
%C A178165 If each element must appear in exactly 1 subset, then we get the Bell numbers A000110.
%C A178165 If each element must appear in exactly 2 subsets, then we get A002718.
%H A178165 Alois P. Heinz, <a href="/A178165/b178165.txt">Table of n, a(n) for n = 0..300</a>
%F A178165 Binomial transform of A094574: a(n) = Sum_{k=0..n} C(n,k)*A094574(k).
%t A178165 terms = m = 30;
%t A178165 a094577[n_] := Sum[Binomial[n, k]*BellB[2n-k], {k, 0, n}];
%t A178165 egf = Exp[(1 - Exp[x])/2]*Sum[a094577[n]*(x/2)^n/n!, {n, 0, m}] + O[x]^m;
%t A178165 A094574 = CoefficientList[egf + O[x]^m, x]*Range[0, m-1]!;
%t A178165 a[n_] := Sum[Binomial[n, k]*A094574[[k+1]], {k, 0, n}];
%t A178165 Table[a[n], {n, 0, m-1}] (* _Jean-François Alcover_, May 24 2019 *)
%o A178165 (Python)
%o A178165 from numpy import array
%o A178165 def toBinary(n, k):
%o A178165     ans=[]
%o A178165     for i in range(k):
%o A178165         ans.insert(0, n%2)
%o A178165         n=n>>1
%o A178165     return array(ans)
%o A178165 def powerSet(k): return [toBinary(n,k) for n in range(1,2**k)]
%o A178165 def courcelle(maxUses, remainingSets, exact=False):
%o A178165     if exact and not all(maxUses<=sum(remainingSets)): ans=0
%o A178165     elif len(remainingSets)==0: ans=1
%o A178165     else:
%o A178165         set0=remainingSets[0]
%o A178165         if all(set0<=maxUses): ans=courcelle(maxUses-set0,remainingSets[1:],exact=exact)
%o A178165         else: ans=0
%o A178165         ans+=courcelle(maxUses,remainingSets[1:],exact=exact)
%o A178165     return ans
%o A178165 for i in range(10):
%o A178165     print(i, courcelle(array([2]*i),powerSet(i),exact=False))
%Y A178165 Row n=2 of A330964.
%Y A178165 Cf. A094574, A000110, A002718, A178171, A178173.
%K A178165 nonn
%O A178165 0,2
%A A178165 _Daniel E. Loeb_, Dec 16 2010
%E A178165 Edited and corrected by _Max Alekseyev_, Dec 19 2010