A190865 -id:A190865 - cp's OEIS frontend

A275422 Number A(n,k) of set partitions of [n] such that k is a multiple of each block size; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 1, 15, 1, 1, 1, 4, 1, 52, 1, 1, 2, 2, 10, 1, 203, 1, 1, 1, 4, 5, 26, 1, 877, 1, 1, 2, 1, 11, 11, 76, 1, 4140, 1, 1, 1, 5, 1, 31, 31, 232, 1, 21147, 1, 1, 2, 1, 14, 2, 106, 106, 764, 1, 115975, 1, 1, 1, 4, 1, 46, 7, 372, 337, 2620, 1, 678570

Offset: 0

Alois P. Heinz, Jul 27 2016

			A(5,3) = 11: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345, 1|2|3|4|5.
A(4,4) = 11: 1234, 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
A(6,5) = 7: 12345|6, 12346|5, 12356|4, 12456|3, 13456|2, 1|23456, 1|2|3|4|5|6.
Square array A(n,k) begins:
:    1, 1,   1,   1,    1,  1,    1, 1,    1, ...
:    1, 1,   1,   1,    1,  1,    1, 1,    1, ...
:    2, 1,   2,   1,    2,  1,    2, 1,    2, ...
:    5, 1,   4,   2,    4,  1,    5, 1,    4, ...
:   15, 1,  10,   5,   11,  1,   14, 1,   11, ...
:   52, 1,  26,  11,   31,  2,   46, 1,   31, ...
:  203, 1,  76,  31,  106,  7,  167, 1,  106, ...
:  877, 1, 232, 106,  372, 22,  659, 2,  372, ...
: 4140, 1, 764, 337, 1499, 57, 2836, 9, 1500, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000110, A000012, A000085, A190865, A190452, A275423, A275424, A275425, A275426, A275427, A275428.

Main diagonal gives A275429.

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, A(n-j, k)*binomial(n-1, j-1)), j=
      `if`(k=0, 1..n, numtheory[divisors](k))))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A[n_, k_] := A[n, k] = If[n==0, 1, Sum[If[j>n, 0, A[n-j, k]*Binomial[n-1, j - 1]], {j, If[k==0, Range[n], Divisors[k]]}]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)

E.g.f. for column k>0: exp(Sum_{d|k} x^d/d!), for k=0: exp(exp(x)-1).

A323293 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have two vertices in common.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 271, 5596, 231577, 21286940, 4392750641, 2100400533176

Offset: 0

Gus Wiseman, Jan 10 2019

			The a(5) = 26 hypergraphs:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,2,5}}
  {{1,3,4}}
  {{1,3,5}}
  {{1,4,5}}
  {{2,3,4}}
  {{2,3,5}}
  {{2,4,5}}
  {{3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
Non-isomorphic representatives of the 6 unlabeled 3-uniform hypertrees spanning 6 vertices where no two edges have two vertices in common, and their multiplicities in the labeled case which add up to a(6) = 271:
    1 X {}
   20 X {{1,2,3}}
   90 X {{1,2,5},{3,4,5}}
   10 X {{1,2,3},{4,5,6}}
  120 X {{1,3,5},{2,3,6},{4,5,6}}
   30 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}

Essentially the same as A287232.

Cf. A000665, A025035, A125791, A190865, A289837, A302374, A302394, A319540, A320395, A322451, A323292-A323299.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]>1&]],{n,8}]

a(9) from Andrew Howroyd, Aug 14 2019

a(10) and a(11) (using A287232) from Joerg Arndt, Oct 12 2023

A362043 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * binomial(n-2j,j)/(n-2j)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 5, 1, 1, 1, 1, 4, 9, 11, 1, 1, 1, 1, 5, 13, 21, 31, 1, 1, 1, 1, 6, 17, 31, 81, 106, 1, 1, 1, 1, 7, 21, 41, 151, 351, 337, 1, 1, 1, 1, 8, 25, 51, 241, 736, 1233, 1205, 1, 1, 1, 1, 9, 29, 61, 351, 1261, 2689, 5769, 5021, 1

Offset: 0

Seiichi Manyama, Apr 15 2023

			Square array begins:
  1,  1,  1,   1,   1,   1,   1, ...
  1,  1,  1,   1,   1,   1,   1, ...
  1,  1,  1,   1,   1,   1,   1, ...
  1,  2,  3,   4,   5,   6,   7, ...
  1,  5,  9,  13,  17,  21,  25, ...
  1, 11, 21,  31,  41,  51,  61, ...
  1, 31, 81, 151, 241, 351, 481, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A000012, A190865, A001470.
Main diagonal gives A362173.
T(n,2*n) gives A362300.
T(n,6*n) gives A362301.
Cf. A359762, A362302.

T(n, k) = n!*sum(j=0, n\3, (k/6)^j/(j!*(n-3*j)!));

E.g.f. of column k: exp(x + k*x^3/6).
T(n,k) = T(n-1,k) + k * binomial(n-1,2) * T(n-3,k) for n > 2.
T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j / (j! * (n-3*j)!).

A323296 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.

Original entry on oeis.org

1, 0, 0, 1, 11, 10, 25, 406, 4823, 15436, 72915, 895180, 11320441, 71777498, 519354927, 6155284240, 82292879425, 788821735656, 7772567489083, 98329764933354, 1400924444610675, 17424772471470490, 216091776292721021, 3035845122991962688, 46700545575567202903

Offset: 0

Gus Wiseman, Jan 11 2019

nonn

The only way to meet the requirements is to cover the vertices with zero or more disconnected 3-uniform hypergraphs with each edge having exactly two vertices in common (A323294). - Andrew Howroyd, Aug 18 2019

			The a(4) = 11:
  {{1,2,3},{1,2,4}}
  {{1,2,3},{1,3,4}}
  {{1,2,3},{2,3,4}}
  {{1,2,4},{1,3,4}}
  {{1,2,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4}}
  {{1,2,3},{1,2,4},{2,3,4}}
  {{1,2,3},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
The following are non-isomorphic representatives of the 3 unlabeled 3-uniform hypergraphs spanning 7 vertices with no two edges having exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 406.
  210 X {{1,2,3},{4,6,7},{5,6,7}}
  140 X {{1,2,3},{4,5,7},{4,6,7},{5,6,7}}
   21 X {{1,6,7},{2,6,7},{3,6,7},{4,6,7},{5,6,7}}
   35 X {{1,2,3},{4,5,6},{4,5,7},{4,6,7},{5,6,7}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323294, A323297.

b:= n-> `if`(n<5, (n-2)*(2*n^2-6*n+3)/6, n/2)*(n-1):
a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, k-1)*b(k)*a(n-k), k=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 18 2019

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]==1&],Union@@#==Range[n]&]],{n,8}]

seq(n)={Vec(serlaplace(exp(-x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x + O(x^(n-1)))/2)))} \\ Andrew Howroyd, Aug 18 2019

From Andrew Howroyd, Aug 18 2019: (Start)
Exponential transform of A323294.
E.g.f.: exp(-x^2/2 - x^3/3 + 5*x^4/24 + x^2*exp(x)/2). (End)

a(11) from Alois P. Heinz, Aug 12 2019

Terms a(12) and beyond from Andrew Howroyd, Aug 18 2019

A323299 Number of 3-uniform hypergraphs on n labeled vertices where every two edges have exactly one vertex in common.

Original entry on oeis.org

1, 1, 1, 2, 5, 26, 261, 3216, 19617, 80860, 262651, 737716, 1920821, 5013152, 14277485, 47610876, 186355041, 820625616, 3869589607, 19039193980, 96332399701, 499138921736, 2639262062801, 14234781051932, 78188865206145, 437305612997376, 2487692697142251

Offset: 0

Gus Wiseman, Jan 11 2019

nonn

			The a(5) = 26 hypergraphs:
  {}
  {{1,2,3}}
  {{1,2,4}}
  {{1,2,5}}
  {{1,3,4}}
  {{1,3,5}}
  {{1,4,5}}
  {{2,3,4}}
  {{2,3,5}}
  {{2,4,5}}
  {{3,4,5}}
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
The following are non-isomorphic representatives of the 10 unlabeled 3-uniform hypergraphs on 7 vertices where every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 3216.
    1 X {}
   35 X {{1,2,3}}
  315 X {{1,2,5},{3,4,5}}
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  840 X {{1,3,5},{2,3,6},{4,5,6}}
  840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}
  630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
   30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}

Cf. A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323296, A323297, A323298.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]!=1&]],{n,8}]

Binomial transform of A323298.

Terms a(11) and beyond from Andrew Howroyd, Aug 14 2019

A362381 E.g.f. satisfies A(x) = exp(x + x^3/6 * A(x)).

Original entry on oeis.org

1, 1, 1, 2, 9, 41, 191, 1191, 9353, 77897, 704861, 7352621, 85323921, 1058023825, 14155416003, 206100005931, 3217934262481, 53320102598481, 939087824434009, 17562552535939705, 346668611080774081, 7196193133818592961, 156944931623033340711

Offset: 0

Seiichi Manyama, Apr 20 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..447
Eric Weisstein's World of Mathematics, Lambert W-Function.

Column k=1 of A362378.

Cf. A190865, A362477.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x-lambertw(-x^3/6*exp(x)))))

E.g.f.: exp(x - LambertW(-x^3/6 * exp(x))) = -6 * LambertW(-x^3/6 * exp(x))/x^3.

a(n) = n! * Sum_{k=0..floor(n/3)} (1/6)^k * (k+1)^(n-2*k-1) / (k! * (n-3*k)!).

A323292 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have two vertices in common.

Original entry on oeis.org

1, 0, 0, 1, 0, 15, 160, 4125, 193200, 19384225

Offset: 0

Gus Wiseman, Jan 10 2019

			The a(5) = 15 hypergraphs:
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
Non-isomorphic representatives of the 3 unlabeled 3-uniform hypergraphs spanning 6 vertices where no two edges have two vertices in common, and their multiplicities in the labeled case which add up to a(6) = 160:
   10 X {{1,2,3},{4,5,6}}
  120 X {{1,3,5},{2,3,6},{4,5,6}}
   30 X {{1,2,4},{1,3,5},{2,3,6},{4,5,6}}

Cf. A000665, A025035, A125791, A190865, A289837, A302374, A302394, A320395, A322451, A323293-A323299.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]>=2&],Union@@#==Range[n]&]],{n,6}]

Inverse binomial transform of A323293. - Andrew Howroyd, Aug 14 2019

a(9) from Andrew Howroyd, Aug 14 2019

A323298 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have exactly one vertex in common.

Original entry on oeis.org

1, 0, 0, 1, 0, 15, 150, 1815, 0, 945, 0, 10395, 0, 135135, 0, 2027025, 0, 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0, 7905853580625, 0, 213458046676875, 0, 6190283353629375, 0, 191898783962510625, 0, 6332659870762850625, 0, 221643095476699771875

Offset: 0

Gus Wiseman, Jan 11 2019

nonn

The only way to cover more than 7 vertices is with edges all having a single common vertex. For the special cases of n = 6 or n = 7, there are also covers without a common vertex. - Andrew Howroyd, Aug 15 2019

			The a(5) = 15 hypergraphs:
  {{1,4,5},{2,3,5}}
  {{1,4,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,4},{2,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,2,5},{3,4,5}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{1,3,4}}
  {{1,2,4},{3,4,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{1,4,5}}
The following are non-isomorphic representatives of the 5 unlabeled 3-uniform hypergraphs spanning 7 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(7) = 1815.
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  840 X {{1,4,5},{2,4,6},{3,4,7},{5,6,7}}
  630 X {{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
  210 X {{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
   30 X {{1,2,7},{1,3,6},{1,4,5},{2,3,5},{2,4,6},{3,4,7},{5,6,7}}
From _Andrew Howroyd_, Aug 15 2019: (Start)
The following are non-isomorphic representatives of the 2 unlabeled 3-uniform hypergraphs spanning 6 vertices in which every two edges have exactly one vertex in common, and their multiplicities in the labeled case, which add up to a(6) = 150.
    120 X {{1,2,3},{1,4,5},{3,5,6}}
     30 X {{1,2,3},{1,4,5},{3,5,6},{2,4,6}}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A001147, A025035, A125791, A190865, A289837, A299471, A302374, A302394, A322451, A323293, A323296, A323297, A323299.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{3}],Length[Intersection[#1,#2]]!=1&],Union@@#==Range[n]&]],{n,10}]

a(n)={if(n%2, if(n<=3, n==3, if(n==7, 1815, n!/(2^(n\2)*(n\2)!))), if(n==6, 150, n==0))} \\ Andrew Howroyd, Aug 15 2019

a(2*n) = 0 for n > 3; a(2*n-1) = A001147(n) for n > 4. - Andrew Howroyd, Aug 15 2019

Terms a(13) and beyond from Andrew Howroyd, Aug 15 2019

A351929 Expansion of e.g.f. exp(x - x^3/6).

Original entry on oeis.org

1, 1, 1, 0, -3, -9, -9, 36, 225, 477, -819, -10944, -37179, 16875, 870507, 4253796, 2481921, -101978919, -680495175, -1060229088, 16378166061, 145672249311, 368320357791, -3415036002300, -40270115077983, -141926533828299, 882584266861701, 13970371667206176

Offset: 0

Seiichi Manyama, Feb 26 2022

sign

Cf. A351930, A351931.

Cf. A190865, A246607.

m = 27; Range[0, m]! * CoefficientList[Series[Exp[x - x^3/6], {x, 0, m}], x] (* Amiram Eldar, Feb 26 2022 *)

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x-x^3/6)))

a(n) = n!*sum(k=0, n\3, (-1/3!)^k*binomial(n-2*k, k)/(n-2*k)!);

a(n) = if(n<3, 1, a(n-1)-binomial(n-1, 2)*a(n-3));

a(n) = n! * Sum_{k=0..floor(n/3)} (-1/6)^k * binomial(n-2*k,k)/(n-2*k)!.

a(n) = a(n-1) - binomial(n-1,2) * a(n-3) for n > 2.

A351932 Number of set partitions of [n] such that block sizes are either 1 or 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 36, 106, 442, 1786, 6106, 23596, 120836, 631632, 2854216, 13590396, 81258556, 510768316, 2839808572, 16008902296, 108643656136, 787965516416, 5161270717296, 33513036683512, 253407796702776, 2065728484459576, 15485032349429176, 113510664648701776

Offset: 0

Seiichi Manyama, Feb 26 2022

nonn

Cf. A000085, A190865, A275423, A275425.

Cf. A190452, A190875, A351930.

a:= proc(n) option remember; `if`(n=0, 1,
     `if`(n<4, 0, a(n-4)*binomial(n-1, 3))+a(n-1))
    end:
seq(a(n), n=0..28);  # Alois P. Heinz, Feb 26 2022
seq(round(evalf(hypergeom([-n/4,(1-n)/4,(2-n)/4,(3-n)/4],[],32/3))),n=0..28);  # Karol A. Penson, Jul 28 2023

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x+x^4/4!)))

a(n) = n!*sum(k=0, n\4, 1/4!^k*binomial(n-3*k, k)/(n-3*k)!);

a(n) = if(n<4, 1, a(n-1)+binomial(n-1, 3)*a(n-4));

E.g.f.: exp(x + x^4/24).
a(n) = n! * Sum_{k=0..floor(n/4)} (1/24)^k * binomial(n-3*k,k)/(n-3*k)!.
a(n) = a(n-1) + binomial(n-1,3) * a(n-4) for n > 3.
a(n) = hypergeom([-n/4,(1-n)/4,(2-n)/4,(3-n)/4],[],32/3), Karol A. Penson, Jul 28 2023.

cp's OEIS Frontend

A275422 Number A(n,k) of set partitions of [n] such that k is a multiple of each block size; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A323293 Number of 3-uniform hypergraphs on n labeled vertices where no two edges have two vertices in common.

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A362043 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * binomial(n-2*j,j)/(n-2*j)!.

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A323296 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have exactly one vertex in common.

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A323299 Number of 3-uniform hypergraphs on n labeled vertices where every two edges have exactly one vertex in common.

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A362381 E.g.f. satisfies A(x) = exp(x + x^3/6 * A(x)).

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A323292 Number of 3-uniform hypergraphs spanning n labeled vertices where no two edges have two vertices in common.

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A323298 Number of 3-uniform hypergraphs spanning n labeled vertices where every two edges have exactly one vertex in common.

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A362043 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..floor(n/3)} (k/6)^j * binomial(n-2j,j)/(n-2j)!.