A054780 -id:A054780 - cp's OEIS frontend

A116508 a(n) = C( C(n,2), n).

1, 0, 0, 1, 15, 252, 5005, 116280, 3108105, 94143280, 3190187286, 119653565850, 4922879481520, 220495674290430, 10682005290753420, 556608279578340080, 31044058215401404845, 1845382436487682488000, 116475817125419611477660, 7779819801401934344268210

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Mar 21 2006

a(n) is the number of simple labeled graphs with n nodes and n edges. - Geoffrey Critzer, Nov 02 2014

These graphs are not necessarily covering, but the covering case is A367863, unlabeled A006649, and the unlabeled version is A001434. - Gus Wiseman, Dec 22 2023

			a(5) = C(C(5,2),5) = C(10,5) = 252.

Alois P. Heinz, Table of n, a(n) for n = 0..370

Main diagonal of A084546.
The unlabeled version is A001434, covering case A006649.
The connected case is A057500, unlabeled A001429.
For set-systems we have A136556, covering case A054780.
The covering case is A367863.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A133686 counts graphs satisfying a strict AOC, connected A129271.
A367867 counts graphs contradicting a strict AOC, connected A140638.
Cf. A001187, A003465, A143543, A305000, A367916, A367917.

[0] cat [(Binomial(Binomial(n+2, n), n+2)): n in [0..20]]; // Vincenzo Librandi, Nov 03 2014

a:= n-> binomial(binomial(n, 2), n):
seq(a(n), n=0..20);

nn = 18; f[x_, y_] :=
Sum[(1 + y)^Binomial[n, 2] x^n/n!, {n, 1, nn}]; Table[
n! Coefficient[Series[f[x, y], {x, 0, nn}], x^n y^n], {n, 1, nn}] (* Geoffrey Critzer, Nov 02 2014 *)
Table[Length[Subsets[Subsets[Range[n],{2}],{n}]],{n,0,5}] (* Gus Wiseman, Dec 22 2023 *)
Table[SeriesCoefficient[(1 + x)^(n*(n-1)/2), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 06 2025 *)

from math import comb
def A116508(n): return comb(n*(n-1)>>1,n) # Chai Wah Wu, Jul 02 2024

[(binomial(binomial(n+2,n),n+2)) for n in range(-1, 17)] # Zerinvary Lajos, Nov 30 2009

a(n) ~ exp(n - 2) * n^(n - 1/2) / (sqrt(Pi) * 2^(n + 1/2)). - Vaclav Kotesovec, May 19 2020

a(n) = [x^n] (1+x)^(n*(n-1)/2). - Vaclav Kotesovec, Aug 06 2025

a(0)=1 prepended by Alois P. Heinz, Feb 02 2024

A014068 a(n) = binomial(n*(n+1)/2, n).

Original entry on oeis.org

1, 1, 3, 20, 210, 3003, 54264, 1184040, 30260340, 886163135, 29248649430, 1074082795968, 43430966148115, 1917283000904460, 91748617512913200, 4730523156632595024, 261429178502421685800, 15415916972482007401455, 966121413245991846673830, 64123483527473864490450300

Offset: 0

N. J. A. Sloane

nonn

Product of next n numbers divided by product of first n numbers. E.g., a(4) = (7*8*9*10)/(1*2*3*4)= 210. - Amarnath Murthy, Mar 22 2004

Also the number of labeled loop-graphs with n vertices and n edges. The covering case is A368597. - Gus Wiseman, Jan 25 2024

			From _Gus Wiseman_, Jan 25 2024: (Start)
The a(0) = 1 through a(3) = 20 loop-graph edge-sets (loops shown as singletons):
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,2}}
             {{2},{1,2}}  {{1},{2},{1,3}}
                          {{1},{2},{2,3}}
                          {{1},{3},{1,2}}
                          {{1},{3},{1,3}}
                          {{1},{3},{2,3}}
                          {{2},{3},{1,2}}
                          {{2},{3},{1,3}}
                          {{2},{3},{2,3}}
                          {{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}}
(End)

Harvey P. Dale, Table of n, a(n) for n = 0..370
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Graph Loop.

Cf. A000217, A022915, A135860, A135861, A135862.
Diagonal of A084546.
Without loops we have A116508, covering A367863, unlabeled A006649.
Allowing edges of any positive size gives A136556, covering A054780.
The covering case is A368597.
The unlabeled version is A368598, covering A368599.
The connected case is A368951.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 (shifted left) counts loop-graphs, covering A322661.
A006129 counts covering simple graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
Cf. A000085, A000272, A057500, A333331, A368596, A368730.

[Binomial(Binomial(n+1,2), n): n in [0..40]]; // G. C. Greubel, Feb 19 2022

Binomial[First[#],Last[#]]&/@With[{nn=20},Thread[{Accumulate[ Range[ 0,nn]], Range[ 0,nn]}]] (* Harvey P. Dale, May 27 2014 *)

from math import comb
def A014068(n): return comb(comb(n+1,2),n) # Chai Wah Wu, Jul 14 2024

[(binomial(binomial(n+1, n-1), n)) for n in range(20)] # Zerinvary Lajos, Nov 30 2009

For n >= 1, Product_{k=1..n} a(k) = A022915(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001
For n > 0, a(n) = A022915(n)/A022915(n-1). - Gerald McGarvey, Jul 26 2004
a(n) = binomial(T(n+1), T(n)) where T(n) = the n-th triangular number. - Amarnath Murthy, Jul 14 2005
a(n) = binomial(binomial(n+2, n), n+1) for n >= -1. - Zerinvary Lajos, Nov 30 2009
From Peter Bala, Feb 27 2020: (Start)
a(p) == (p + 1)/2 ( mod p^3 ) for prime p >= 5 (apply Mestrovic, equation 37).
Conjectural: a(2*p) == p*(2*p + 1) ( mod p^4 ) for prime p >= 5. (End)
a(n) = A084546(n,n). - Gus Wiseman, Jan 25 2024
a(n) = [x^n] (1+x)^(n*(n+1)/2). - Vaclav Kotesovec, Aug 06 2025

A136556 a(n) = binomial(2^n - 1, n).

Original entry on oeis.org

1, 1, 3, 35, 1365, 169911, 67945521, 89356415775, 396861704798625, 6098989894499557055, 331001552386330913728641, 64483955378425999076128999167, 45677647585984911164223317311276545, 118839819203635450208125966070067352769535, 1144686912178270649701033287538093722740144666625

Offset: 0

Paul D. Hanna, Jan 07 2008; Paul Hanna and Vladeta Jovovic, Jan 15 2008

nonn

Number of n x n binary matrices without zero rows and with distinct rows up to permutation of rows, cf. A014070.
Row 0 of square array A136555.
From Gus Wiseman, Dec 19 2023: (Start)
Also the number of n-element sets of nonempty subsets of {1..n}, or set-systems with n vertices and n edges (not necessarily covering). The covering case is A054780. For example, the a(3) = 35 set-systems are:
  {1}{2}{3}  {1}{2}{12}  {1}{2}{123}  {1}{12}{123}  {12}{13}{123}
             {1}{2}{13}  {1}{3}{123}  {1}{13}{123}  {12}{23}{123}
             {1}{2}{23}  {1}{12}{13}  {1}{23}{123}  {13}{23}{123}
             {1}{3}{12}  {1}{12}{23}  {2}{12}{123}
             {1}{3}{13}  {1}{13}{23}  {2}{13}{123}
             {1}{3}{23}  {2}{3}{123}  {2}{23}{123}
             {2}{3}{12}  {2}{12}{13}  {3}{12}{123}
             {2}{3}{13}  {2}{12}{23}  {3}{13}{123}
             {2}{3}{23}  {2}{13}{23}  {3}{23}{123}
                         {3}{12}{13}  {12}{13}{23}
                         {3}{12}{23}
                         {3}{13}{23}
Of these, only {{1},{2},{1,2}}, {{1},{3},{1,3}}, and {{2},{3},{2,3}} do not cover the vertex set.
(End)

			G.f.: A(x) = 1 + x + 3*x^2 + 35*x^3 + 1365*x^4 + 169911*x^5 +...
A(x) = 1/(1+x) + log(1+2*x)/(1+2*x) + log(1+4*x)^2/(2!*(1+4*x)) + log(1+8*x)^3/(3!*(1+8*x)) + log(1+16*x)^4/(4!*(1+16*x)) + log(1+32*x)^5/(5!*(1+32*x)) +...

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

Sequences of the form binomial(2^n +p*n +q, n): this sequence (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
Cf. A066384, A136555, A136557.
The covering case A054780 has binomial transform A367916, ranks A367917.
Connected graphs of this type are A057500, unlabeled A001429.
Graphs of this type are A116508, covering A367863, unlabeled A006649.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A058891 counts set-systems, connected A323818, without singletons A016031.
Cf. A055154, A059201, A133686, A367862, A367902, A367903.

[Binomial(2^n -1, n): n in [0..20]]; // G. C. Greubel, Mar 14 2021

A136556:= n-> binomial(2^n-1,n); seq(A136556(n), n=0..20); # G. C. Greubel, Mar 14 2021

f[n_] := Binomial[2^n - 1, n]; Array[f, 12] (* Robert G. Wilson v *)
Table[Length[Subsets[Rest[Subsets[Range[n]]],{n}]],{n,0,4}] (* Gus Wiseman, Dec 19 2023 *)

{a(n) = binomial(2^n-1,n)}
for(n=0, 20, print1(a(n), ", "))

/* As coefficient of x^n in the g.f.: */
{a(n) = polcoeff( sum(i=0,n, 1/(1 + 2^i*x +x*O(x^n)) * log(1 + 2^i*x +x*O(x^n))^i/i!), n)}
for(n=0, 20, print1(a(n), ", "))

from math import comb
def A136556(n): return comb((1<Chai Wah Wu, Jan 02 2024

[binomial(2^n -1, n) for n in (0..20)] # G. C. Greubel, Mar 14 2021

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2^n,k).
a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k) * (2^n-1)^k.
G.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n! * (1 + 2^n*x)).
a(n) ~ 2^(n^2)/n!. - Vaclav Kotesovec, Jul 02 2016

Edited by N. J. A. Sloane, Jan 26 2008

A368597 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges.

Original entry on oeis.org

1, 1, 3, 17, 150, 1803, 27364, 501015, 10736010, 263461265, 7283725704, 223967628066, 7581128184175, 280103206674480, 11216492736563655, 483875783716549277, 22371631078155742764, 1103548801569848115255, 57849356643299101021960, 3211439288584038922502820

Offset: 0

Gus Wiseman, Jan 04 2024

nonn

It doesn't matter for this sequence whether we use loops such as {x,x} or half-loops such as {x}.

			The a(0) = 1 through a(3) = 17 set-systems:
  {}  {{1}}  {{1},{2}}    {{1},{2},{3}}
             {{1},{1,2}}  {{1},{2},{1,3}}
             {{2},{1,2}}  {{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}}

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

This is the covering case of A014068.
Allowing edges of any positive size gives A054780, covering case of A136556.
Allowing any number of edges gives A322661, connected A062740.
The case of just pairs is A367863, covering case of A116508.
The unlabeled version is A368599.
The version contradicting strict AOC is A368730.
The connected case is A368951.
A000085 counts set partitions into singletons or pairs.
A006129 counts covering graphs, connected A001187.
A058891 counts set-systems, unlabeled A000612.
A100861 counts set partitions into singletons or pairs by number of pairs.
A111924 counts set partitions into singletons or pairs by length.
Cf. A000272, A000666, A057500, A066383, A333331, A367869, A368596, A368600, A368928, A368951, A369199.

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

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n)) \\ Andrew Howroyd, Jan 06 2024

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * binomial(binomial(k+1,2), n). - Andrew Howroyd, Jan 06 2024

Terms a(7) and beyond from Andrew Howroyd, Jan 06 2024

A116532 Number of singular n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

Original entry on oeis.org

0, 0, 3, 285, 50820, 23551920, 31898503077, 134251404794199

Offset: 1

Vladeta Jovovic, Apr 03 2006

Cf. A000410, A116506, A116527.

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

a(n) = A054780(n) - A088389(n).

A055154 Triangle read by rows: T(n,k) = number of k-covers of a labeled n-set, k=1..2^n-1.

Original entry on oeis.org

1, 1, 3, 1, 1, 12, 32, 35, 21, 7, 1, 1, 39, 321, 1225, 2919, 4977, 6431, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 1, 120, 2560, 24990, 155106, 711326, 2597410, 7856550, 20135050, 44337150, 84665490, 141118250, 206252550, 265182450, 300540190

Offset: 1

Vladeta Jovovic, Jun 14 2000

Row sums give A003465.
From Manfred Boergens, Apr 11 2024: (Start)
If more than half of the nonempty subsets of [n] are drawn their union covers [n] (see Formula). - The proof is based on 2^(n-1)-1 being the number of nonempty subsets of [n] with one fixed element of [n] missing.
For covers which may include one empty set see A163353.
For disjoint covers see A008277.
For disjoint covers which may include one empty set see A256894 (amendment by Manfred Boergens, Mar 09 2025). (End)

			Triangle begins:
  [1],
  [1,3,1],
  [1,12,32,35,21,7,1],
  ...
There are 35 4-covers of a labeled 3-set.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 165.

Alois P. Heinz, Rows n = 1..10, flattened

Cf. A054780, A055621.
Cf. A369950 (partial row sums).
Cf. 256894.
Columns: A029858, A095152, A095153, A095155.

nn=5;Map[Select[#,#>0&]&,Transpose[Table[Table[Sum[(-1)^j Binomial[n,j] Binomial[2^(n-j)-1,m],{j,0,n}],{n,1,nn}],{m,1,2^nn-1}]]]//Grid (* Geoffrey Critzer, Jun 27 2013 *)

T(n,k) = Sum_{j=0..n} (-1)^j*C(n, j)*C(2^(n-j)-1, k), k=1..2^n-1.
From Vladeta Jovovic, May 30 2004: (Start)
T(n,k) = (1/k!)*Sum_{j=0..k} Stirling1(k+1, j+1)*(2^j-1)^n.
E.g.f.: Sum(exp(y*(2^n-1))*log(1+x)^n/n!, n=0..infinity)/(1+x). (End)
Also exp(-y)*Sum((1+x)^(2^n-1)*y^n/n!, n=0..infinity).
From Manfred Boergens, Apr 11 2024: (Start)
T(n,k) = C(2^n-1,k) for k>=2^(n-1).
T(n,k) < C(2^n-1,k) for k<2^(n-1).
(Note: C(2^n-1,k) is the number of all k-subsets of P([n])\{{}}.) (End)

A367916 Number of sets of nonempty subsets of {1..n} with the same number of edges as covered vertices.

Original entry on oeis.org

1, 2, 6, 45, 1376, 161587, 64552473, 85987037645, 386933032425826, 6005080379837219319, 328011924848834642962619, 64153024576968812343635391868, 45547297603829979923254392040011994, 118654043008142499115765307533395739785599

Offset: 0

Gus Wiseman, Dec 08 2023

nonn

			The a(0) = 1 through a(2) = 6 set-systems:
  {}  {}     {}
      {{1}}  {{1}}
             {{2}}
             {{1},{2}}
             {{1},{1,2}}
             {{2},{1,2}}

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

The covering case is A054780.
For graphs we have A367862, covering A367863, unlabeled A006649.
These set-systems have ranks A367917.
A000372 counts antichains, covering A006126, nonempty A014466.
A003465 counts set-systems covering {1..n}, unlabeled A055621.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A136556 counts set-systems on {1..n} with n edges.
Cf. A092918, A102896, A133686, A306445, A323818, A355740, A367770, A367869, A367901, A367902, A367905.

Table[Length[Select[Subsets[Rest[Subsets[Range[n]]]], Length[Union@@#]==Length[#]&]],{n,0,3}]

\\ Here b(n) is A054780(n).
b(n) = sum(k=0, n, (-1)^(n-k) * binomial(n,k) * binomial(2^k-1, n))
a(n) = sum(k=0, n, binomial(n,k) * b(k)) \\ Andrew Howroyd, Dec 29 2023

Binomial transform of A054780.

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

Original entry on oeis.org

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[#]]&]

A368186 Number of n-covers of an unlabeled n-set.

Original entry on oeis.org

1, 1, 2, 9, 87, 1973, 118827, 20576251, 10810818595, 17821875542809, 94589477627232498, 1651805220868992729874, 96651473179540769701281003, 19238331716776641088273777321428, 13192673305726630096303157068241728202, 31503323006770789288222386469635474844616195

Offset: 0

Gus Wiseman, Dec 19 2023

nonn

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 set-systems:
  {{1}}  {{1},{2}}    {{1},{2},{3}}
         {{1},{1,2}}  {{1},{2},{1,3}}
                      {{1},{1,2},{1,3}}
                      {{1},{1,2},{2,3}}
                      {{1},{2},{1,2,3}}
                      {{1},{1,2},{1,2,3}}
                      {{1,2},{1,3},{2,3}}
                      {{1},{2,3},{1,2,3}}
                      {{1,2},{1,3},{1,2,3}}

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

The labeled version is A054780, ranks A367917, non-covering A367916.
The case of graphs is A006649, labeled A367863, cf. A116508, A367862.
The case of connected graphs is A001429, labeled A057500.
Covers with any number of edges are counted by A003465, 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, A002494, A006126, A055130, A133686, A140638, A305000, A317795, A326754, A367901, A367902, A367903.

brute[m_]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i, p[[i]]},{i,Length[p]}])], {p,Permutations[Union@@m]}];
Table[Length[Union[First[Sort[brute[#]]]& /@ Select[Subsets[Rest[Subsets[Range[n]]],{n}], Union@@#==Range[n]&]]], {n,0,3}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={2^sum(j=1, #q, gcd(t, q[j])) - 1}
G(n,m)={if(n==0, 1, my(s=0); forpart(q=n, my(g=sum(t=1, m, K(q,t)*x^t/t, O(x*x^m))); s+=permcount(q)*exp(g - subst(g,x,x^2))); s/n!)}
a(n)=if(n ==0, 1, polcoef(G(n,n) - G(n-1,n), n)) \\ Andrew Howroyd, Jan 03 2024

a(n) = A055130(n, n) for n > 0. - Andrew Howroyd, Jan 03 2024

Terms a(6) and beyond from Andrew Howroyd, Jan 03 2024

cp's OEIS Frontend

A116508 a(n) = C( C(n,2), n).

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A014068 a(n) = binomial(n*(n+1)/2, n).

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A136556 a(n) = binomial(2^n - 1, n).

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A368597 Number of n-element sets of singletons or pairs of distinct elements of {1..n} with union {1..n}, or loop-graphs covering n vertices with n edges.

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A116532 Number of singular n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

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A055154 Triangle read by rows: T(n,k) = number of k-covers of a labeled n-set, k=1..2^n-1.

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