A322706 - OEIS

A322706

Regular triangle read by rows where T(n,k) is the number of k-regular k-uniform hypergraphs spanning n vertices.

1, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 12, 12, 1, 0, 1, 70, 330, 70, 1, 0, 1, 465, 11205, 11205, 465, 1, 0, 1, 3507, 505505, 2531200, 505505, 3507, 1, 0

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OFFSET

1,8

COMMENTS

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is k-uniform if all edges contain exactly k vertices, and k-regular if all vertices belong to exactly k edges. The span of a hypergraph is the union of its edges.

LINKS

Table of n, a(n) for n=1..36.

EXAMPLE

Triangle begins:

1 0

1 1 0

1 3 1 0

1 12 12 1 0

1 70 330 70 1 0

1 465 11205 11205 465 1 0

1 3507 505505 2531200 505505 3507 1 0

Row 4 counts the following hypergraphs:

{{1}{2}{3}{4}} {{12}{13}{24}{34}} {{123}{124}{134}{234}}

{{12}{14}{23}{34}}

{{13}{14}{23}{24}}

MATHEMATICA

Table[Table[SeriesCoefficient[Product[1+Times@@x/@s, {s, Subsets[Range[n], {k}]}], Sequence@@Table[{x[i], 0, k}, {i, n}]], {k, 1, n}], {n, 1, 6}]

CROSSREFS

Row sums are A322705. Second column is A001205. Third column is A110101.

Cf. A005176, A058891, A059441, A295193, A306021, A319056, A319189, A319190, A319612, A321721, A322704.

Sequence in context: A034374 A261318 A103879 * A051722 A357354 A166408

Adjacent sequences: A322703 A322704 A322705 * A322707 A322708 A322709

KEYWORD

nonn,more,tabl

AUTHOR

Gus Wiseman, Dec 23 2018

STATUS

approved