A054548 -id:A054548 - cp's OEIS frontend

A276639 Triangle T(m, n) = the number of point-labeled graphs with n points and m edges, no points isolated. By rows, n >= 0, ceiling(n/2) <= m <= binomial(n,2).

1, 1, 3, 1, 3, 16, 15, 6, 1, 30, 135, 222, 205, 120, 45, 10, 1, 15, 330, 1581, 3760, 5715, 6165, 4945, 2997, 1365, 455, 105, 15, 1, 315, 4410, 23604, 73755, 159390, 259105, 331716, 343161, 290745, 202755, 116175, 54257, 20349, 5985, 1330, 210, 21, 1

Offset: 1

David Pasino, Sep 08 2016

The row sums are A006129, omitting row 1 and A006129(1).

			Triangle T(n, m) begins:
n/m  0  1  2  3   4    5    6    7    8    9   10
0    1  0  0  0   0    0    0    0    0    0   0
1    0  0  0  0   0    0    0    0    0    0   0
2    0  1  0  0   0    0    0    0    0    0   0
3    0  0  3  1   0    0    0    0    0    0   0
4    0  0  3  16  15   6    1    0    0    0   0
5    0  0  0  30  135  222  205  120  45   10  1

David Pasino, Table of n, a(n) for n = 1..512

Another version is A054548.

Table[Sum[Binomial[n, k] (-1)^(n - k) Binomial[Binomial[k, 2], m], {k, 0, n}], {n, 7}, {m, Ceiling[n/2], Binomial[n, 2]}] /. {} -> {1} // Flatten (* Michael De Vlieger, Sep 19 2016 *)

T(n, m) = Sum_{k=0,..n} binomial(n, k) * (-1)^(n-k) * A084546(k, m).

A369198 Number of labeled loop-graphs with n vertices and the same number of edges as covered vertices.

Original entry on oeis.org

1, 2, 6, 30, 241, 2759, 40824, 736342, 15622835, 380668095, 10467815086, 320529284621, 10813165015074, 398413594789777, 15917197015926392, 685312404706694574, 31631317971844128229, 1558017329350990780607, 81567807853701988869120, 4522975947689168088308305

Offset: 0

Gus Wiseman, Jan 18 2024

nonn

			The a(0) = 1 through a(3) = 30 loop-graphs (loops shown as singletons):
  {}  {}     {}           {}
      {{1}}  {{1}}        {{1}}
             {{2}}        {{2}}
             {{1},{2}}    {{3}}
             {{1},{1,2}}  {{1},{2}}
             {{2},{1,2}}  {{1},{3}}
                          {{2},{3}}
                          {{1},{1,2}}
                          {{1},{1,3}}
                          {{2},{1,2}}
                          {{2},{2,3}}
                          {{3},{1,3}}
                          {{3},{2,3}}
                          {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1},{1,3},{2,3}}
                          {{2},{1,2},{1,3}}
                          {{2},{1,2},{2,3}}
                          {{2},{1,3},{2,3}}
                          {{3},{1,2},{1,3}}
                          {{3},{1,2},{2,3}}
                          {{3},{1,3},{2,3}}
                          {{1,2},{1,3},{2,3}}

The version counting all vertices is A014068.
The loopless case is A367862, counting all vertices A116508.
The covering case is A368597, connected A368951.
With inequality we have A369196, covering A369194, connected A369197.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000272, A000666, A006649, A062740, A066383, A367863, A369192, A369193.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Length[#]==Length[Union@@#]&]],{n,0,5}]

Binomial transform of A368597.

A386399 Number of forests with at most n unlabeled nodes.

Original entry on oeis.org

1, 2, 4, 7, 13, 23, 43, 80, 156, 309, 638, 1348, 2949, 6607, 15206, 35720, 85625, 208588, 515787, 1291316, 3269194, 8355832, 21539988, 55942920, 146271594, 384746580, 1017522228, 2704227858, 7219183490, 19351410860, 52068524665, 140588391713, 380824067016

Offset: 0

Max Alekseyev, Jul 20 2025

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Partial sums of A005195.

Cf. A000272, A002807, A006897, A054548, A137917, A367863, A372176.

G.f.: exp(sum_{k>0} B(x^k)/k ) / (1-x), where B(x) = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 11*x^7 + ... = C(x)-1 and C is the g.f. for A000055.

cp's OEIS Frontend

A276639 Triangle T(m, n) = the number of point-labeled graphs with n points and m edges, no points isolated. By rows, n >= 0, ceiling(n/2) <= m <= binomial(n,2).

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A369198 Number of labeled loop-graphs with n vertices and the same number of edges as covered vertices.

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Mathematica

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A386399 Number of forests with at most n unlabeled nodes.

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