A136556 -id:A136556 - cp's OEIS frontend

A368951 Number of connected labeled graphs with n edges and n vertices and with loops allowed.

1, 1, 2, 10, 79, 847, 11436, 185944, 3533720, 76826061, 1880107840, 51139278646, 1530376944768, 49965900317755, 1767387701671424, 67325805434672100, 2747849045156064256, 119626103584870552921, 5533218319763109888000, 270982462739224265922466

Offset: 0

Andrew Howroyd, Jan 10 2024

nonn

Exponential transform appears to be A333331. - Gus Wiseman, Feb 12 2024

			From _Gus Wiseman_, Feb 12 2024: (Start)
The a(0) = 1 through a(3) = 10 loop-graphs:
  {}  {11}  {11,12}  {11,12,13}
            {22,12}  {11,12,23}
                     {11,13,23}
                     {22,12,13}
                     {22,12,23}
                     {22,13,23}
                     {33,12,13}
                     {33,12,23}
                     {33,13,23}
                     {12,13,23}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A000169, A333331, A063170, A053506.
This is the connected covering case of A014068.
The case without loops is A057500, covering case of A370317.
Allowing any number of edges gives A062740, connected case of A322661.
This is the connected case of A368597.
The unlabeled version is A368983, connected case of A368984.
For at most n edges we have A369197.
A000085 counts set partitions into singletons or pairs.
A006129 counts covering graphs, connected A001187.
Cf. A000272, A000666, A054780, A116508, A136556, A367863, A368600.

egf:= (L-> 1-L/2-log(1+L)/2-L^2/4)(LambertW(-x)):
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 10 2024

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(-log(1-t)/2 + t/2 - t^2/4 + 1))}

a(n) = A000169(n) + A057500(n) for n > 0.
E.g.f.: 1 - log(1-T(x))/2 + T(x)/2 - T(x)^2/4 where T(x) = -LambertW(-x) is the e.g.f. of A000169.
From Peter Luschny, Jan 10 2024: (Start)
a(n) = (exp(n)*Gamma(n + 1, n) - (n - 1)*n^(n - 1))/(2*n) for n > 0.
a(n) = (1/2)*(A063170(n)/n - A053506(n)) for n > 0.  (End)

A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 10, 11, 13, 14, 17, 19, 21, 22, 24, 26, 28, 34, 35, 37, 38, 40, 41, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 128, 129, 130, 131, 133, 134, 136, 137, 138, 139, 141, 142, 145, 147, 149, 150, 152

Offset: 1

Gus Wiseman, Dec 12 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. A set-system is a finite set of finite nonempty sets. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

			The terms together with the corresponding set-systems begin:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  17: {{1},{1,3}}
  19: {{1},{2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  24: {{3},{1,3}}
  26: {{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  34: {{2},{2,3}}
  35: {{1},{2},{2,3}}
  37: {{1},{1,2},{2,3}}

These set-systems are counted by A054780 and A367916, A368186.
Graphs of this type are A367862, covering A367863, unlabeled A006649.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, connected A323818, unlabeled A000612.
A070939 gives length of binary expansion.
A136556 counts set-systems on {1..n} with n edges.
Cf. A057500, A059201, A072639, A096111, A116508, A309326, A326031, A326702, A326753, A326754, A367770, A367902, A367905.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]],1];
Select[Range[0,100], Length[bpe[#]]==Length[Union@@bpe/@bpe[#]]&]

A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

Original entry on oeis.org

1, 1, 2, 10, 97, 2160, 126862, 21485262, 11105374322, 18109358131513, 95465831661532570, 1660400673336788987026, 96929369602251313489896310, 19268528295096123543660356281600, 13203875101002459910158494602665950757, 31517691852305548841992346407978317698725021

Offset: 0

Gus Wiseman, Jan 07 2024

nonn

			Non-isomorphic representatives of the a(3) = 10 set-systems:
  {{1},{2},{3}}
  {{1},{2},{1,2}}
  {{1},{2},{1,3}}
  {{1},{2},{1,2,3}}
  {{1},{1,2},{1,3}}
  {{1},{1,2},{2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{1,2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The case of graphs is A001434, labeled A116508.
Labeled version is A136556, covering A054780, binomial transform of A367916.
The case of labeled covering graphs is A367863, binomial transform A367862.
These include the set-systems ranked by A367917.
The covering case is A368186, for graphs A006649, connected A057500.
Requiring all edges to be singletons or pairs gives A368598.
A003465 counts covers with any number of edges, unlabeled A055621.
A046165 counts minimal covers, ranks A309326.
A058891 counts set-systems, unlabeled A000612, without singletons A016031.
A059201 counts covering T_0 set-systems, unlabeled A319637, ranks A326947.
Cf. A000088, A007716, A283877, A367901, A367902, A367903, A368599.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Subsets[Subsets[Range[n],{1,n}],{n}]]],{n,0,4}]

a(n) = polcoef(G(n, n), n) \\ G defined in A368186. - Andrew Howroyd, Jan 11 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 11 2024

A136557 a(n) = Sum_{k=0..n} binomial(2^k + n-k-1, k).

Original entry on oeis.org

1, 2, 6, 45, 1436, 171836, 68149425, 89431630740, 396956313475102, 6099399658235428041, 331007760926212498510464, 64484289650612910347505873728, 45677712418497545460138258802186905

Offset: 0

Paul D. Hanna, Jan 07 2008

nonn

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A136555, A136556.

[(&+[Binomial(2^k +n-k-1, k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Mar 15 2021

A136557:= n-> add(binomial(2^k +n-k-1, k), k=0..n); seq(A136557(n), n=0..20); # G. C. Greubel, Mar 15 2021

Table[Sum[Binomial[2^k+n-k-1, k], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

PARI
```
a(n)=sum(k=0,n,binomial(2^k+n-k-1,k))
```

/* As coefficient of x^n in the g.f.: */ {a(n)=polcoeff(sum(i=0,n,((1+2^i*x+x*O(x^n))*(1-x-2^i*x^2))^-1*log(1+2^i*x+x*O(x^n))^i/i!),n)}

[sum(binomial(2^k +n-k-1, k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Mar 15 2021

Equals antidiagonal sums of square array A136555.
G.f.: A(x) = Sum_{n>=0} (1+2^n*x)^-1 * (1-x-2^n*x^2)^-1 * log(1+2^n*x)^n / n!.
a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

A246699 Squarefree n such that C(2^n - 1, n) is also squarefree, where C is the binomial coefficient.

Original entry on oeis.org

1, 2, 3, 6, 11, 21, 29, 31, 51, 55, 57

Offset: 1

Juri-Stepan Gerasimov, Nov 15 2014

nonn

Conjectured to be finite.

The subsequence of squarefree numbers in A245569. - M. F. Hasler, Nov 28 2014

Cf. A000225, A005117, A136556.

[n: n in [1..200] | IsSquarefree(n) and IsSquarefree(Binomial(2^n-1, n))];

Select[Range[100], SquareFreeQ[#] && SquareFreeQ[Binomial[2^# - 1, #]] &] (* Vincenzo Librandi, Nov 14 2014 *)
Select[Range[60],AllTrue[{#,Binomial[2^#-1,#]},SquareFreeQ]&] (* Harvey P. Dale, Feb 07 2025 *)

is(n)=issquarefree(n) && issquarefree(binomial(2^n-1,n)) \\ Charles R Greathouse IV, Nov 16 2014

A245569 Numbers n such that binomial(2^n-1,n) is squarefree.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 11, 12, 21, 28, 29, 31, 51, 54, 55, 57

Offset: 1

M. F. Hasler, Nov 17 2014

Motivated by the previously existing entry for the subsequence A246699 of squarefree terms in this sequence.

Conjectured to be finite. Next term, if it exists, is larger than 500.

Cf. A000225, A005117, A136556.

Select[Range[0,57],SquareFreeQ[Binomial[2^#-1,#]]&] (* Ivan N. Ianakiev, Nov 18 2014 *)

PARI
```
is(n)=issquarefree(binomial(2^n-1,n))
```

is(n)={t=1; n&&n=vecsort(concat(vector(n,k,factor(if(kx)(t=(2^n-k)/k/denominator(t)),0)~))); for(i=2,#n,if(n[1,i]==n[1,i-1],n[2,i]+=n[2,i-1];n[2,i-1]=0));!n||vecmax(n[2,])<2} \\ Limiting "factor" to precomputed primes might yield a false positive, but no term > 57 is found so far.

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A368951 Number of connected labeled graphs with n edges and n vertices and with loops allowed.

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A367917 BII-numbers of set-systems with the same number of edges as covered vertices.

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A368731 Number of non-isomorphic n-element sets of nonempty subsets of {1..n}.

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A136557 a(n) = Sum_{k=0..n} binomial(2^k + n-k-1, k).

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A246699 Squarefree n such that C(2^n - 1, n) is also squarefree, where C is the binomial coefficient.

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A245569 Numbers n such that binomial(2^n-1,n) is squarefree.

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