A001862 -id:A001862 - cp's OEIS frontend

A066383 a(n) = Sum_{k=0..n} C(n*(n+1)/2,k).

1, 2, 7, 42, 386, 4944, 82160, 1683218, 40999516, 1156626990, 37060382822, 1328700402564, 52676695500313, 2287415069586304, 107943308165833912, 5499354613856855290, 300788453960472434648, 17577197510240126035698, 1092833166733915284972350

Offset: 0

N. J. A. Sloane, Dec 23 2001

nonn

Number of labeled loop-graphs with n vertices and at most n edges. - Gus Wiseman, Feb 14 2024

			From _Gus Wiseman_, Feb 14 2024: (Start)
The a(0) = 1 through a(2) = 7 loop-graphs (loops shown as singletons):
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}
(End)

Harry J. Smith, Table of n, a(n) for n = 0..100

The case of equality is A014068, covering A368597.
The loopless version is A369192, covering A369191.
The covering case is A369194, minimal case A001862.
Counting only covered vertices gives A369196, without loops A369193.
The connected covering case is A369197, without loops A129271.
The unlabeled version is A370168, covering A370169.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000085, A000666, A005703, A062740, A116508, A367862, A367863, A367916, A368927, A369141, A369199.

f[n_] := Sum[Binomial[n (n + 1)/2, k], {k, 0, n}]; Array[f, 21, 0] (* Vincenzo Librandi, May 06 2016 *)
Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]],Length[#]<=n&]],{n,0,5}] (* Gus Wiseman, Feb 14 2024 *)

{ for (n=0, 100, a=0; for (k=0, n, a+=binomial(n*(n + 1)/2, k)); write("b066383.txt", n, " ", a) ) } \\ Harry J. Smith, Feb 12 2010

from math import comb
def A066383(n): return sum(comb(comb(n+1,2),k) for k in range(n+1)) # Chai Wah Wu, Jul 10 2024

a(n) = 2^(n*(n+1)/2) - binomial(n*(n+1)/2,n+1)*2F1(1,(-n^2+n+2)/2;n+2;-1) = A006125(n) - A116508(n+1) * 2F1(1,(-n^2+n+2)2;n+2;-1), where 2F1(a,b;c;x) is the hypergeometric function. - Ilya Gutkovskiy, May 06 2016

a(n) ~ exp(n) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, Feb 20 2024

A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

Original entry on oeis.org

1, 1, 4, 23, 199, 2313, 34015, 606407, 12712643, 306407645, 8346154699, 253476928293, 8490863621050, 310937199521774, 12356288017546937, 529516578044589407, 24339848939829286381, 1194495870124420574751, 62332449791125883072149, 3446265450868329833016605

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

Row-sums of left portion of A369199.

			The a(0) = 1 through a(3) = 23 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{2},{1,3}}
             {{1},{1,2}}  {{3},{1,2}}
             {{2},{1,2}}  {{1,2},{1,3}}
                          {{1,2},{2,3}}
                          {{1},{2},{3}}
                          {{1,3},{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 minimal case is A001862, without loops A053530.
This is the covering case of A066383 and A369196, cf. A369192 and A369193.
The case of equality is A368597, without loops A367863.
The version without loops is A369191.
The connected case is A369197, without loops A129271.
The unlabeled version is A370169, equality A368599, non-covering A368598.
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.
A133686 counts choosable graphs, covering A367869.
A322661 counts covering loop-graphs, unlabeled A322700.
A367867 counts non-choosable graphs, covering A367868.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000272, A000666, A003465, A005703, A006649, A057500, A062740, A116508, A367862, A367916.

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

Inverse binomial transform of A369196.

A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

Original entry on oeis.org

1, 2, 7, 39, 320, 3584, 51405, 900947, 18661186, 445827942, 12062839691, 364451604095, 12157649050827, 443713171974080, 17583351295466338, 751745326170662049, 34485624653535808340, 1689485711682987916502, 88030098291829749593643, 4860631073631586486397141

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 1 through a(2) = 7 loop-graphs:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1,2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

The version counting all vertices is A066383, without loops A369192.
The loopless case is A369193, with case of equality A367862.
The covering case is A369194, connected A369197, minimal case A001862.
The case of equality is A369198, covering case A368597.
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. A000169, A000272, A000666, A001187, A006649, A057500, A062740, A116508, A367863, A369145.

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

Binomial transform of A369194.

A370317 Number of labeled graphs with n vertices (allowing isolated vertices) and n edges, such that the edge set is connected.

Original entry on oeis.org

1, 0, 0, 1, 15, 252, 4905, 110715, 2864148, 83838720, 2744568522, 99463408335, 3955626143040, 171344363805582, 8031863998136355, 405150528051451000, 21884686370917378050, 1260420510502767861840, 77105349570138633021624, 4993117552678619556356085

Offset: 0

Gus Wiseman, Feb 17 2024

nonn

			The a(0) = 0 through a(4) = 15 graphs:
  {}  .  .  {{1,2},{1,3},{2,3}}  {{1,2},{1,3},{1,4},{2,3}}
                                 {{1,2},{1,3},{1,4},{2,4}}
                                 {{1,2},{1,3},{1,4},{3,4}}
                                 {{1,2},{1,3},{2,3},{2,4}}
                                 {{1,2},{1,3},{2,3},{3,4}}
                                 {{1,2},{1,3},{2,4},{3,4}}
                                 {{1,2},{1,4},{2,3},{2,4}}
                                 {{1,2},{1,4},{2,3},{3,4}}
                                 {{1,2},{1,4},{2,4},{3,4}}
                                 {{1,2},{2,3},{2,4},{3,4}}
                                 {{1,3},{1,4},{2,3},{2,4}}
                                 {{1,3},{1,4},{2,3},{3,4}}
                                 {{1,3},{1,4},{2,4},{3,4}}
                                 {{1,3},{2,3},{2,4},{3,4}}
                                 {{1,4},{2,3},{2,4},{3,4}}

Andrew Howroyd, Table of n, a(n) for n = 0..100

The covering case is A057500.
This is the connected case of A116508.
Allowing any number of edges gives A287689.
Counting only covered vertices gives A370318.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, connected A001187.
A369192 counts graphs with at most n edges, covering A369191.
Cf. A000169, A001862, A054780, A062740, A129271, A367863, A369193, A369197.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Length[#]==n&&Length[csm[#]]<=1&]], {n,0,5}]

a(n)=n!*polcoef(polcoef(exp(x + O(x*x^n))*(1 + log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k,2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024

a(n) = n!*[x^n][y^n] exp(x)*(1 + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Feb 19 2024

a(8) onwards from Andrew Howroyd, Feb 19 2024

A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 0, 0, 3, 10, 12, 6, 1, 0, 0, 0, 16, 79, 162, 179, 116, 45, 10, 1, 0, 0, 0, 0, 125, 847, 2565, 4615, 5540, 4720, 2948, 1360, 455, 105, 15, 1, 0, 0, 0, 0, 0, 1296, 11436, 47100, 121185, 220075, 301818, 325578, 282835, 200115, 115560, 54168, 20343, 5985, 1330, 210, 21, 1

Offset: 0

Gus Wiseman, Jan 19 2024

This sequence excludes the graph consisting of a single isolated vertex without a loop. - Andrew Howroyd, Feb 02 2024

			Triangle begins:
    1
    0    1
    0    1    2    1
    0    0    3   10   12    6    1
    0    0    0   16   79  162  179  116   45   10    1
Row n = 3 counts the following loop-graphs (loops shown as singletons):
  .  .  {12,13}  {1,12,13}   {1,2,12,13}   {1,2,3,12,13}   {1,2,3,12,13,23}
        {12,23}  {1,12,23}   {1,2,12,23}   {1,2,3,12,23}
        {13,23}  {1,13,23}   {1,2,13,23}   {1,2,3,13,23}
                 {2,12,13}   {1,3,12,13}   {1,2,12,13,23}
                 {2,12,23}   {1,3,12,23}   {1,3,12,13,23}
                 {2,13,23}   {1,3,13,23}   {2,3,12,13,23}
                 {3,12,13}   {1,12,13,23}
                 {3,12,23}   {2,3,12,13}
                 {3,13,23}   {2,3,12,23}
                 {12,13,23}  {2,3,13,23}
                             {2,12,13,23}
                             {3,12,13,23}

Andrew Howroyd, Table of n, a(n) for n = 0..1560 (rows 0..20)

Row lengths are A000124.
Diagonal T(n,n-1) is A000272, rooted A000169.
The case without loops is A062734.
Row sums are A062740.
Transpose is A322147.
Column sums are A322151.
Diagonal T(n,n) is A368951, connected case of A368597.
Connected case of A369199, without loops A054548.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts simple graphs, also loop-graphs if shifted left.
A006129 counts covering graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A001862, A014068, A054923, A057500, A066383, A322114, A322137, A369197.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,2}],{k}], Length[Union@@#]==n&&Length[csm[#]]<=1&]], {n,0,5},{k,0,Binomial[n+1,2]}]

T(n)={[Vecrev(p) | p<-Vec(serlaplace(1 - x + log(sum(j=0, n, (1 + y)^binomial(j+1, 2)*x^j/j!, O(x*x^n))))) ]}
{ my(A=T(6)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Feb 02 2024

E.g.f.: 1 - x + log(Sum_{j >= 0} (1 + y)^binomial(j+1, 2)*x^j/j!). - Andrew Howroyd, Feb 02 2024

cp's OEIS Frontend

A066383 a(n) = Sum_{k=0..n} C(n*(n+1)/2,k).

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A369194 Number of labeled loop-graphs covering n vertices with at most n edges.

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A369196 Number of labeled loop-graphs with n vertices and at most as many edges as covered vertices.

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A370317 Number of labeled graphs with n vertices (allowing isolated vertices) and n edges, such that the edge set is connected.

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A369195 Irregular triangle read by rows where T(n,k) is the number of labeled connected loop-graphs covering n vertices with k edges.

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