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A306020 a(n) is the number of set-systems using nonempty subsets of {1,...,n} in which all sets have the same size.

%I A306020 #24 Apr 02 2025 10:19:53
%S A306020 1,2,5,16,95,2110,1114237,68723671292,1180735735906024030715,
%T A306020 170141183460507917357914971986913657850,
%U A306020 7237005577335553223087828975127304179197147198604070555943173844710572689401
%N A306020 a(n) is the number of set-systems using nonempty subsets of {1,...,n} in which all sets have the same size.
%C A306020 A058673(n) <= a(n). - _Lorenzo Sauras Altuzarra_, Aug 10 2023
%H A306020 <a href="/index/Mat#matroid">Index entries for sequences related to matroids</a>
%F A306020 a(n) = 1 - n + Sum_{d = 1..n} 2^binomial(n, d).
%e A306020 a(3) = 16 set-systems in which all sets have the same size:
%e A306020   {}
%e A306020   {{1}}
%e A306020   {{2}}
%e A306020   {{3}}
%e A306020   {{1,2}}
%e A306020   {{1,3}}
%e A306020   {{2,3}}
%e A306020   {{1,2,3}}
%e A306020   {{1},{2}}
%e A306020   {{1},{3}}
%e A306020   {{2},{3}}
%e A306020   {{1,2},{1,3}}
%e A306020   {{1,2},{2,3}}
%e A306020   {{1,3},{2,3}}
%e A306020   {{1},{2},{3}}
%e A306020   {{1,2},{1,3},{2,3}}
%p A306020 a := n -> 1-n+add(2^binomial(n, d), d = 1 .. n):
%p A306020 seq(a(n), n = 0 .. 10); # _Lorenzo Sauras Altuzarra_, Aug 11 2023
%t A306020 Table[1+Sum[2^Binomial[n,d]-1,{d,n}],{n,10}]
%o A306020 (PARI) a(n) = 1 - n + sum(d = 1, n, 2^binomial(n, d)); \\ _Michel Marcus_, Aug 10 2023
%Y A306020 Cf. A000005, A001315, A007716, A038041, A049311, A058673 (labeled matroids), A283877, A298422, A306017, A306018, A306019, A306021.
%Y A306020 Row sums of A059328.
%K A306020 nonn
%O A306020 0,2
%A A306020 _Gus Wiseman_, Jun 17 2018