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A299471 Regular triangle where T(n,k) is the number of labeled k-uniform hypergraphs spanning n vertices.

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