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%I #21 Jan 16 2024 22:05:53
%S 1,1,1,1,4,1,1,41,11,1,1,768,958,26,1,1,27449,1042642,32596,57,1,1,
%T 1887284,34352419335,34359509614,2096731,120,1,1,252522481,
%U 72057319189324805,1180591620442534312297,72057594021152435,268434467,247,1,1,66376424160
%N Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.
%H Andrew Howroyd, <a href="/A299471/b299471.txt">Table of n, a(n) for n = 1..91</a> (rows 1..13)
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>
%F T(n, k) = Sum_{d = 0..n} (-1)^(n-d)*binomial(n,d)*2^binomial(d,k).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 41, 11, 1;
%e 1, 768, 958, 26, 1;
%e 1, 27449, 1042642, 32596, 57, 1;
%e ...
%t Table[Sum[(-1)^(n-d)*Binomial[n,d]*2^Binomial[d,k],{d,0,n}],{n,10},{k,n}]
%o (PARI) T(n, k) = sum(d = 0, n, (-1)^(n-d)*binomial(n,d)*2^binomial(d,k)) \\ _Andrew Howroyd_, Jan 16 2024
%Y Columns 1..4 are A000012, A006129, A302374, A302396.
%Y Row sums are A306021.
%Y The unlabeled version is A301922.
%Y The connected version is A299354.
%Y Cf. A000005, A001315, A006126, A038041, A298422, A298426, A306017, A306018, A306019, A306020.
%K nonn,tabl
%O 1,5
%A _Gus Wiseman_, Jun 18 2018
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