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A306021 Number of set-systems spanning {1,...,n} in which all sets have the same size.

%I A306021 #22 Jan 16 2024 22:05:39
%S A306021 1,1,2,6,54,1754,1102746,68715913086,1180735735356265746734,
%T A306021 170141183460507906731293351306656207090,
%U A306021 7237005577335553223087828975127304177495735363998991435497132232365910414322
%N A306021 Number of set-systems spanning {1,...,n} in which all sets have the same size.
%C A306021 a(n) is the number of labeled uniform hypergraphs spanning n vertices. - _Andrew Howroyd_, Jan 16 2024
%H A306021 Andrew Howroyd, <a href="/A306021/b306021.txt">Table of n, a(n) for n = 0..14</a>
%F A306021 a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(1 - k + Sum_{d = 1..k} 2^binomial(k, d)).
%F A306021 Inverse binomial transform of A306020. - _Andrew Howroyd_, Jan 16 2024
%e A306021 The a(3) = 6 set-systems in which all sets have the same size:
%e A306021   {{1,2,3}}
%e A306021   {{1}, {2}, {3}}
%e A306021   {{1,2}, {1,3}}
%e A306021   {{1,2}, {2,3}}
%e A306021   {{1,3}, {2,3}}
%e A306021   {{1,2}, {1,3}, {2,3}}
%t A306021 Table[Sum[(-1)^(n-k)*Binomial[n,k]*(1+Sum[2^Binomial[k,d]-1,{d,k}]),{k,0,n}],{n,12}]
%o A306021 (PARI) a(n) = if(n==0, 1, sum(k=0, n, sum(d=0, n, (-1)^(n-d)*binomial(n,d)*2^binomial(d,k)))) \\ _Andrew Howroyd_, Jan 16 2024
%Y A306021 Row sums of A299471.
%Y A306021 The unlabeled version is A301481.
%Y A306021 The connected version is A299353.
%Y A306021 Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306017, A306018, A306019, A306020.
%K A306021 nonn
%O A306021 0,3
%A A306021 _Gus Wiseman_, Jun 17 2018