A035053 -id:A035053 - cp's OEIS frontend

A144977 a(n) = A134955(n) - A134955(n-2).

1, 1, 3, 7, 16, 41, 108, 301, 881, 2684, 8455, 27444, 91248, 309593, 1068584, 3742171, 13269281, 47561455, 172092274, 627887239, 2307902495, 8539497952, 31786480760, 118960956585, 447413177185, 1690336204778, 6412656031161

Offset: 1

Washington Bomfim, Sep 28 2008

nonn

a(n) is the number of hyperforests with n unlabeled nodes without trees of order 2. This follows from the fact that for n >= 2 A134955(n-2) counts the hyperforests of order n with one or more trees of order 2.
The unique hyperforest (without loops) of order 1 is an isolated vertex, so a(1) = 1.
For n >= 2, a(n) - a(n-1) counts hyperforests of order n with components of order >= 3.

			a(3) = 3 since the only options are 2 hypertrees of order 3, or the forest composed by 3 isolated nodes.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Cf. A134955, A035053 (hypertrees).

\\ here b(n) is A007563 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); v}
seq(n)={my(u=b(n)); my(v=Vec(Ser(EulerT(u))*(1-x*Ser(u)))); EulerT(vector(#v, n, if(n<>2, v[n], 0)))} \\ Andrew Howroyd, Aug 27 2018

A304977 Number of unlabeled hyperforests spanning n vertices with singleton edges allowed.

Original entry on oeis.org

1, 1, 4, 14, 55, 235, 1112, 5672, 30783, 175733, 1042812, 6385278, 40093375, 257031667, 1676581863, 11098295287, 74401300872, 504290610004, 3451219615401, 23821766422463, 165684694539918, 1160267446543182, 8175446407807625, 57928670942338011, 412561582740147643

Offset: 0

Gus Wiseman, May 22 2018

nonn

			Non-isomorphic representatives of the a(3) = 14 hyperforests are the following:
  {{1,2,3}}
  {{3},{1,2}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{2},{3},{1,3}}
  {{2},{3},{1,2,3}}
  {{3},{1,2},{2,3}}
  {{3},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{2},{3},{1,2},{1,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}

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

Cf. A030019, A035053, A134954, A134955, A134956, A134957, A134958, A134959, A144959, A304867, A304911, A304912, A304918.

\\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); concat([1], EulerT(Vec(Ser(EulerT(u))*(1-x*Ser(u))-1)))} \\ Andrew Howroyd, Aug 27 2018

Euler transform of b(1) = 1, b(n > 1) = A134959(n).

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

A144979 Number of hyperforests on n unlabeled nodes, assuming that each edge contains at least two nodes, with all components of prime orders.

Original entry on oeis.org

0, 1, 2, 3, 11, 15, 70, 92, 166, 351, 5061, 5782, 60736, 73183, 135152, 303426, 8507114, 9468630, 119603007, 140712654, 262160102, 593434948, 21042972101, 23146479248, 44736887989, 96738104613, 122459045525

Offset: 1

Washington Bomfim, Sep 28 2008

nonn

			a(5) = 11 since the only options are 9 hypertrees of order 5, or the two hyperforests composed by components of order 3 and 2.

D. E. Knuth: The Art of Computer Programming, Volume 4, Generating All Combinations and Partitions Fascicle 3, Section 7.2.1.4. Generating all partitions. Page 38, Algorithm H.

Cf. A035053 (hypertrees), A000040 (prime numbers).

a(n) = Sum of prod_{k=1..n} C(A035053(k)+c_k-1,c_k}) over the partitions of n having all parts k prime, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0.

A303674 Number of connected integer partitions of n > 1 whose distinct parts are pairwise indivisible and whose z-density is -1.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 4, 1, 6, 4, 5, 1, 8, 2, 7, 5, 11, 3, 11, 5, 13, 6, 14, 7, 19, 6, 19, 15, 24, 13, 28, 15, 33, 20, 34, 22, 46, 30, 48, 32, 57, 39, 67, 48, 76, 63, 88, 62, 104, 88, 110, 94, 130, 115, 164, 121, 172, 152, 198, 176, 229, 203, 270, 235, 293, 272, 341, 311, 375, 349, 453, 420, 506, 452, 570, 547

Offset: 1

Gus Wiseman, Jun 04 2018

nonn

The z-density of a multiset S is defined to be Sum_{s in S} (omega(s) - 1) - omega(lcm(S)), where omega = A001221 and lcm is least common multiple.

Given a finite multiset S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.

			The a(18) = 8 integer partitions are (18), (14,4), (10,8), (9,9), (10,4,4), (6,4,4,4), (3,3,3,3,3,3), (2,2,2,2,2,2,2,2,2).
The a(20) = 7 integer partitions are (20), (14,6), (12,8), (10,6,4), (5,5,5,5), (4,4,4,4,4), (2,2,2,2,2,2,2,2,2,2).

Cf. A030019, A035053, A048143, A134954, A286520, A293510, A293994, A303837, A303838, A304382.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[IntegerPartitions[n],And[zensity[#]==-1,Length[zsm[#]]==1,Select[Tuples[#,2],UnsameQ@@#&&Divisible@@#&]=={}]&]],{n,30}]

a(51)-a(81) from Robert Price, Sep 15 2018

A304919 Number of labeled hyperforests spanning {1,...,n} and allowing singleton edges.

Original entry on oeis.org

1, 1, 5, 45, 665, 14153, 399421, 14137301, 603647601, 30231588689, 1738713049013, 112976375651901, 8186616300733321, 654642360222892057, 57267075701210437229, 5440407421313402397541, 557802495215406348358113, 61393838258161429159571873, 7220049654850517272144419941, 903546142463635579042416518989

Offset: 0

Gus Wiseman, May 21 2018

nonn

			The a(2) = 5 hyperforests are the following:
{{1,2}}
{{1},{2}}
{{1},{1,2}}
{{2},{1,2}}
{{1},{2},{1,2}}

Cf. A030019, A035053, A048143, A134954, A134955, A134956, A134957, A134959, A144959, A303838, A304717, A304867, A304911, A304912, A304918.

Inverse binomial transform of A134956.

cp's OEIS Frontend

A144977 a(n) = A134955(n) - A134955(n-2).

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A304977 Number of unlabeled hyperforests spanning n vertices with singleton edges allowed.

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A144979 Number of hyperforests on n unlabeled nodes, assuming that each edge contains at least two nodes, with all components of prime orders.

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A303674 Number of connected integer partitions of n > 1 whose distinct parts are pairwise indivisible and whose z-density is -1.

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A304919 Number of labeled hyperforests spanning {1,...,n} and allowing singleton edges.

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