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 A306021 Number of set-systems spanning {1,...,n} in which all sets have the same size. 43
 1, 1, 2, 6, 54, 1754, 1102746, 68715913086, 1180735735356265746734, 170141183460507906731293351306656207090, 7237005577335553223087828975127304177495735363998991435497132232365910414322 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) is the number of labeled uniform hypergraphs spanning n vertices. - Andrew Howroyd, Jan 16 2024 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..14 FORMULA a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*(1 - k + Sum_{d = 1..k} 2^binomial(k, d)). Inverse binomial transform of A306020. - Andrew Howroyd, Jan 16 2024 EXAMPLE The a(3) = 6 set-systems in which all sets have the same size: {{1,2,3}} {{1}, {2}, {3}} {{1,2}, {1,3}} {{1,2}, {2,3}} {{1,3}, {2,3}} {{1,2}, {1,3}, {2,3}} MATHEMATICA Table[Sum[(-1)^(n-k)*Binomial[n, k]*(1+Sum[2^Binomial[k, d]-1, {d, k}]), {k, 0, n}], {n, 12}] PROG (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 CROSSREFS Row sums of A299471. The unlabeled version is A301481. The connected version is A299353. Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306017, A306018, A306019, A306020. Sequence in context: A085078 A152543 A279454 * A122593 A267348 A264610 Adjacent sequences: A306018 A306019 A306020 * A306022 A306023 A306024 KEYWORD nonn AUTHOR Gus Wiseman, Jun 17 2018 STATUS approved

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Last modified September 14 13:32 EDT 2024. Contains 375921 sequences. (Running on oeis4.)