cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A367752 Number of shapes of labeled rooted hypertrees with n vertices.

Original entry on oeis.org

1, 1, 4, 29, 256, 3007, 42932, 721121, 13982563, 306967231, 7527903208, 203977383469, 6051630040496, 195111205542541, 6792697846367791, 253966747582533681, 10149075292428481965, 431705938073882999275, 19474660918369182445456, 928660364396786865580881
Offset: 1

Views

Author

Paul Laubie, Nov 29 2023

Keywords

Comments

The shape of a labeled rooted hypertree is a labeled rooted hypertrees where we replace all the maximal subtrees by a corolla rooted on a new unlabeled black vertex.
If we remove the black vertices that are the parent of only 1 white vertex, we obtain labeled rooted hypertrees with black and white vertices such that:
- black vertices are unlabeled;
- black vertices have at least two children;
- the children of a black vertex are white, and are connected to it via simple edges (edges connecting only two vertices);
- the children of a white vertex are connected to it via hyperedges (edges connecting strictly more than two vertices).

Examples

			For n = 3 the a(3) = 4 solutions are:
  - the corolla with a black root which have 3 white children,
  - and the 3 possible labeling of the hypertree with a white root which have 2 white children connected to it via a hyperedge.

Crossrefs

Cf. A052888, A210586, A364709, A364816.

Programs

  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(serreverse(log(1+x)*exp(-exp(x)+x+1)))) \\ Michel Marcus, Nov 30 2023
  • SageMath
    R.=PowerSeriesRing(QQ);(ln(1+t)*exp(-exp(t)+t+1)).reverse().egf_to_ogf().list()[1:]

Formula

E.g.f.: series reversion of log(1+x)*exp(-exp(x)+x+1).

A367753 Number of shapes of labeled forest of rooted hypertrees with n vertices.

Original entry on oeis.org

1, 2, 8, 55, 507, 5969, 85605, 1445420, 28110615, 618760615, 15207174501, 412790636977, 12265337498000, 395962288858946, 13800754780797740, 516494067220932259, 20658199248901273576, 879406216174705907137, 39698270229941320201019, 1894212537494300993244732
Offset: 1

Views

Author

Paul Laubie, Nov 29 2023

Keywords

Comments

See A367752 for the definition of the shape of a labeled rooted hypertree.

Crossrefs

Cf. A052888, A210586, A364709, A364816, A367752.

Programs

  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(serreverse((1+x)*log(1+log(1+x))*exp(-x)))) \\ Michel Marcus, Nov 30 2023
  • SageMath
    R.=PowerSeriesRing(QQ,30);(1+t)*(ln(1+ln(1+t))*exp(-t)).reverse().egf_to_ogf().list()[1:]

Formula

E.g.f.: series reversion of (1+x)*log(1+log(1+x))*exp(-x).

A370948 Triangle read by rows: T(n,k) is the number of labeled forests of rooted Greg hypertrees with n white vertices and weight k, 0 <= k < n.

Original entry on oeis.org

1, 3, 1, 22, 15, 1, 262, 271, 53, 1, 4336, 6020, 2085, 165, 1, 91984, 160336, 81310, 13040, 487, 1, 2381408, 4996572, 3364011, 851690, 73024, 1407, 1, 72800928, 178613156, 150499951, 53119521, 7696794, 383649, 4041, 1
Offset: 1

Views

Author

Paul Laubie, Mar 06 2024

Keywords

Comments

A rooted Greg hypertree is a hypertree with black and white vertices such that white vertices are labeled, black vertices are unlabeled, and each black vertex has at least two children.
The weight of a forest of rooted Greg hypertrees is the number of hypertrees minus 1 plus the weight of each hyperedge which is the number of vertices it connects minus 2. See A364709 for the analog sequence for hypertrees. A forest of rooted Greg hypertrees of weight 0 is exactly a Greg tree.

Examples

			Triangle T(n,k) begins:
n\k    0     1     2     3     4 ...
1      1;
2      3,    1;
3     22,   15,    1;
4    262,  271,   53,    1;
5   4336, 6020, 2085,  165,    1;
...

Links

Crossrefs

Cf. A364709, A005264 (k=0), A370949.
Row sums are A364816.
Series reversion as e.g.f. is related to A092271.

Programs

  • PARI
    T(n)={my(x='x+O('x^(n+1))); [Vecrev(p) | p <- Vec(serlaplace(serreverse( (log(1+y*x)/y - exp(x) + x + 1)*exp(-x) )))]}
{ my(A=T(8)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Mar 06 2024

Formula

E.g.f: series reversion in t of (log(1+v*t)/v - exp(t) + t + 1)*exp(-t), where the formal variable v encodes the weight.
T(n,0) = A005264(n).
T(n,n-1) = 1.
Showing 1-3 of 3 results.

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