A293510 -id:A293510 - cp's OEIS frontend

A323818 Number of connected set-systems covering n vertices.

1, 1, 4, 96, 31840, 2147156736, 9223372011084915712, 170141183460469231602560095199828453376, 57896044618658097711785492504343953923912733397452774312021795134847892828160

Offset: 0

Gus Wiseman, Jan 30 2019

nonn

Unlike the nearly identical sequence A092918, this sequence does not count under a(1) the a single-vertex hypergraph with no edges.

			The a(2) = 4 set-systems:
  {{1, 2}}
  {{1}, {1,2}}
  {{2}, {1,2}}
  {{1}, {2}, {1,2}}

G. C. Greubel, Table of n, a(n) for n = 0..11

Cf. A001187, A003465 (not necessarily connected), A016031, A048143, A092918, A293510, A317672, A323816, A323817 (no singletons), A323819 (unlabeled case).

m:=12;
f:= func< x | 1-x + Log( (&+[2^(2^n-1)*x^n/Factorial(n): n in [0..m+2]]) ) >;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Oct 04 2022

b:= n-> add(binomial(n, k)*2^(2^(n-k)-1)*(-1)^k, k=0..n):
a:= proc(n) option remember; b(n)-`if`(n=0, 0, add(
       k*binomial(n, k)*b(n-k)*a(k), k=1..n-1)/n)
    end:
seq(a(n), n=0..8);  # Alois P. Heinz, Jan 30 2019

nn=8;
ser=Sum[2^(2^n-1)*x^n/n!,{n,0,nn}];
Table[SeriesCoefficient[1-x+Log[ser],{x,0,n}]*n!,{n,0,nn}]

m=12;
def f(x): return 1-x + log(sum(2^(2^n-1)*x^n/factorial(n) for n in range(m+2)))
def A_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A_list(m) # G. C. Greubel, Oct 04 2022

E.g.f.: 1 - x + log(Sum_{n >= 0} 2^(2^n-1) * x^n/n!).

Logarithmic transform of A003465.

A007297 Number of connected graphs on n labeled nodes on a circle with straight-line edges that don't cross.

Original entry on oeis.org

1, 1, 4, 23, 156, 1162, 9192, 75819, 644908, 5616182, 49826712, 448771622, 4092553752, 37714212564, 350658882768, 3285490743987, 30989950019532, 294031964658430, 2804331954047160, 26870823304476690, 258548658860327880

Offset: 1

N. J. A. Sloane, Mira Bernstein

Apart from the initial 1, reversion of g.f. for A162395 (squares with signs): see A263843.

			G.f. = x*(1 + x + 4*x^2 + 23*x^3 + 156*x^4 + 1162*x^5 + 9192*x^6 + 75819*x^7 + ...).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..999 (first 101 terms from T. D. Noe)
M. Aguiar and J.-L. Loday, Quadri-algebras, J. Pure Appl. Algebra, 191 (2004), 205-221.
R. Bacher, On generating series of complementary plane trees arXiv:math/0409050 [math.CO], 2004.
P. Barry and A. Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From _N. J. A. Sloane_, Sep 21 2012
F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.
C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358 (column sums in Table 2).
Michael Drmota, Anna de Mier, Marc Noy, Extremal statistics on non-crossing configurations, Discrete Math. 327 (2014), 103--117. MR3192420. See Eq. (3). - N. J. A. Sloane, May 18 2014
Guillermo Esteban, Clemens Huemer, Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, On the congruences of some combinatorial numbers, Stud. Appl. Math. vol. 116 (2006) pp. 135-144.
P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486.
Loïc Foissy, Free quadri-algebras and dual quadri-algebras, arXiv:1504.06056 [math.CO], 2015.
I. M. Gessel, A short proof of the Deutsch-Sagan congruence for connected non crossing graphs, arXiv preprint arXiv:1403.7656 [math.CO], 2014.
Marco Kuhlmann, Tabulation of Noncrossing Acyclic Digraphs, arXiv:1504.04993 [cs.DS], 2015.
M. Kuhlmann, P. Jonsson, Parsing to Noncrossing Dependency Graphs, Transactions of the Association for Computational Linguistics, vol. 3, pp. 559-570, 2015.
Sara Madariaga, Gröbner-Shirshov bases for the non-symmetric operads of dendriform algebras and quadri-algebras, arXiv:1304.5184 [math.RA], 2013.
B. Vallette, Manin products, Koszul duality, Loday algebras and Deligne conjecture, arXiv:math/0609002 [math.QA], 2006-2007; J. Reine Angew. Math. 620 (2008), 105-164.
Gus Wiseman, The a(5) = 156 connected non-crossing graphs.
Gus Wiseman, Constructive Mathematica program for A007297.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
Index entries for reversions of series

Cf. A162395, A000290. 4th row of A107111. Row sums of A089434.
See A263843 for a variant.
Cf. A000108 (non-crossing set partitions), A001006, A001187, A054726 (non-crossing graphs), A054921, A099947, A194560, A293510, A323818, A324167, A324169, A324173.

A007297:=proc(n) if n = 1 then 1 else add(binomial(3*n - 3, n + j)*binomial(j - 1, j - n + 1), j = n - 1 .. 2*n - 3)/(n - 1); fi; end;

CoefficientList[ InverseSeries[ Series[(x-x^2)/(1+x)^3, {x, 0, 20}], x], x] // Rest (* From Jean-François Alcover, May 19 2011, after PARI prog. *)
Table[Binomial[3n, 2n+1] Hypergeometric2F1[1-n, n, 2n+2, -1]/n, {n, 1, 20}] (* Vladimir Reshetnikov, Oct 25 2015 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x)^3+O(x^(n+2))),n+1)) /* Ralf Stephan */

Apart from initial term, g.f. is the series reversion of (x-x^2)/(1+x)^3 (A162395). See A263843. - Vladimir Kruchinin, Feb 08 2013
G.f.: (g-z)/z, where g=-1/3+(2/3)*sqrt(1+9z)*sin((1/3)*arcsin((2+27z+54z^2)/2/(1+9*z)^(3/2))). - Emeric Deutsch, Dec 02 2002
a(n) = (1/n)*Sum_{k=0..n} binomial(3n, n-k-1)*binomial(n+k-1, k). - Paul Barry, May 11 2005
a(n) = 4^(n-1)*(Gamma(3*n/2-1)/Gamma(n/2+1)/Gamma(n) -Gamma((3*n-1)/2)/ Gamma( (n+1)/2)/Gamma(n+1)). - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar
a(n) = 4^n * binomial(3*n/2, n/2) / (9*n-6) - 4^(n-1) * binomial(3*(n-1)/2, (n-1)/2 ) / n. - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar
D-finite with recurrence: n*(n-1)*(3*n-4)*a(n) +36*(n-1)*a(n-1) -12*(3*n-8)*(3*n-1)*(3*n-7)*a(n-2)=0. - Mark van Hoeij, Aug 27 2005, adapted to offset Feb 21 2020 by R. J. Mathar
a(n) = (1/n)*Sum_{k=0..n} C(3n, k)*C(2n-k-2, n-1). - Paul Barry, Sep 27 2005
a(n) ~ (2-sqrt(3)) * 6^n * 3^(n/2) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2014
a(n) = binomial(3*n,2*n+1)*hypergeom([1-n,n], [2*n+2], -1)/n. - Vladimir Reshetnikov, Oct 25 2015
a(n) = 2*A078531(n) - A085614(n+1). - Vladimir Reshetnikov, Apr 24 2016

Better description from Philippe Flajolet, Apr 20 2000
More terms from James Sellers, Aug 21 2000
Definition revised and initial a(1)=1 added by N. J. A. Sloane, Nov 05 2015 at the suggestion of Axel Boldt. Some of the formulas may now need to be adjusted slightly.

A326754 BII-numbers of set-systems covering an initial interval of positive integers.

Original entry on oeis.org

0, 1, 3, 4, 5, 6, 7, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 26, 27, 28, 29, 30, 31, 33, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78

Offset: 1

Gus Wiseman, Jul 23 2019

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. 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. Elements of a set-system are sometimes called edges.

			The sequence of all covering set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   3: {{1},{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   7: {{1},{2},{1,2}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  13: {{1},{1,2},{3}}
  14: {{2},{1,2},{3}}
  15: {{1},{2},{1,2},{3}}
  18: {{2},{1,3}}
  19: {{1},{2},{1,3}}
  20: {{1,2},{1,3}}
  21: {{1},{1,2},{1,3}}
  22: {{2},{1,2},{1,3}}
  23: {{1},{2},{1,2},{1,3}}
  26: {{2},{3},{1,3}}
  27: {{1},{2},{3},{1,3}}
  28: {{1,2},{3},{1,3}}
  29: {{1},{1,2},{3},{1,3}}
  30: {{2},{1,2},{3},{1,3}}

John Tyler Rascoe, Table of n, a(n) for n = 1..10000

Cf. A000120, A003465, A006126, A048793, A070939, A293510, A326031, A326702.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[0,100],normQ[Join@@bpe/@bpe[#]]&]

from itertools import chain, count, islice
def bin_i(n): #binary indices
    return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
def a_gen():
    for n in count(0):
        s = set(i for i in chain.from_iterable([bin_i(k) for k in bin_i(n)]))
        y = len(s)
        if sum(s) == (y*(y+1))//2:
            yield n
A326754_list = list(islice(a_gen(), 100)) # John Tyler Rascoe, Jun 20 2024

A324173 Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 2, 6, 6, 1, 0, 6, 15, 20, 10, 1, 0, 21, 51, 65, 50, 15, 1, 0, 85, 203, 252, 210, 105, 21, 1, 0, 385, 912, 1120, 938, 560, 196, 28, 1, 0, 1907, 4527, 5520, 4620, 2898, 1302, 336, 36, 1, 0, 10205, 24370, 29700, 24780, 15792, 7812, 2730, 540, 45, 1

Offset: 0

Gus Wiseman, Feb 17 2019

A set partition is crossing if it contains a pair of blocks of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

The topologically connected components of a set partition correspond to the blocks of its minimal non-crossing coarsening.

			Triangle begins:
     1
     0     1
     0     1     1
     0     1     3     1
     0     2     6     6     1
     0     6    15    20    10     1
     0    21    51    65    50    15     1
     0    85   203   252   210   105    21     1
     0   385   912  1120   938   560   196    28     1
     0  1907  4527  5520  4620  2898  1302   336    36     1
     0 10205 24370 29700 24780 15792  7812  2730   540    45     1
Row n = 4 counts the following set partitions:
  {{1234}}    {{1}{234}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
  {{13}{24}}  {{12}{34}}  {{1}{23}{4}}
              {{123}{4}}  {{12}{3}{4}}
              {{124}{3}}  {{1}{24}{3}}
              {{134}{2}}  {{13}{2}{4}}
              {{14}{23}}  {{14}{2}{3}}

FindStat, The number of topologically connected components of a set partition.

Row sums are A000110. Column k = 1 is A099947.
Cf. A000108, A001263, A002061, A002662, A007297, A016098, A048993, A054726, A293510, A305078, A305079, A323818.
Cf. A324166, A324167, A324169, A324170, A324171, A324172.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Length[crosscmpts[#]]==k&]],{n,0,8},{k,0,n}]

A309326 BII-numbers of minimal covers.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 320, 512, 513, 516, 520, 521, 524, 528, 544, 545, 548

Offset: 1

Gus Wiseman, Jul 23 2019

nonn

Elements of a set-system are sometimes called edges. A minimal cover is a set-system where every edge contains at least one vertex that does not belong to any other edge.

			The sequence of all minimal covers together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    3: {{1},{2}}
    4: {{1,2}}
    8: {{3}}
    9: {{1},{3}}
   10: {{2},{3}}
   11: {{1},{2},{3}}
   12: {{1,2},{3}}
   16: {{1,3}}
   18: {{2},{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   33: {{1},{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  129: {{1},{4}}

Cf. A000120, A003465, A006126, A048793, A070939, A293510, A326031, A326702.

Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326749 (connected), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,1000],And@@Table[Union@@Delete[bpe/@bpe[#],i]!=Union@@bpe/@bpe[#],{i,Length[bpe/@bpe[#]]}]&]

A317672 Regular triangle where T(n,k) is the number of clutters (connected antichains) on n + 1 vertices with k maximal blobs (2-connected components).

Original entry on oeis.org

1, 2, 3, 44, 24, 16, 4983, 940, 300, 125, 7565342, 154770, 18000, 4320, 1296, 2414249587694, 318926314, 3927105, 363580, 72030, 16807, 56130437054842366160898, 135200580256336, 10244647168, 99187200, 8028160, 1376256, 262144

Offset: 1

Gus Wiseman, Aug 03 2018

			Triangle begins:
        1
        2       3
       44      24      16
     4983     940     300     125
  7565342  154770   18000    4320    1296

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Row sums are A048143. First column is A275307. Last column is A030019.

Cf. A000272, A001187, A002218, A013922, A134954, A293510, A317631, A317632, A317634, A317635, A317671.

blg={0,1,2,44,4983,7565342,2414249587694,56130437054842366160898} (* A275307 *);
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[n^(k-1)*Product[blg[[Length[s]+1]],{s,spn}],{spn,Select[sps[Range[n-1]],Length[#]==k&]}],{n,Length[blg]},{k,n-1}]

A323819 Number of non-isomorphic connected set-systems covering n vertices.

Original entry on oeis.org

1, 1, 3, 30, 1912, 18662590, 12813206131799685, 33758171486592987138461432668177794, 1435913805026242504952006868879460423767388571975632398910903473535427583

Offset: 0

Gus Wiseman, Jan 30 2019

nonn

			Non-isomorphic representatives of the a(3) = 30 set-systems:
  {{1,2,3}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{3},{1,3},{2,3}}
  {{1},{2,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{3},{2,3},{1,2,3}}
  {{2},{3},{2,3},{1,2,3}}
  {{3},{1,2},{1,3},{2,3}}
  {{2},{1,3},{2,3},{1,2,3}}
  {{3},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{2,3},{1,2,3}}
  {{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{1,3},{2,3},{1,2,3}}
  {{2},{3},{1,3},{2,3},{1,2,3}}
  {{3},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3},{1,2,3}}
  {{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

Alois P. Heinz, Table of n, a(n) for n = 0..12
Thomas Delacroix, Meaningful objective frequency-based interesting pattern mining, Thesis, 2021.
Geon Lee, Seokbum Yoon, Jihoon Ko, Hyunju Kim, and Kijung Shin, Hypergraph Motifs and Their Extensions Beyond Binary, arXiv:2310.15668 [cs.SI], 2023.

Cf. A000295, A003465, A016031, A048143, A055621 (not necessarily connected), A293510, A317795, A323817, A323818 (labeled case).

nmax = 12;
b[n_, i_, l_] := b[n, i, l] = If[n == 0, 2^Function[w, Sum[Product[2^GCD[t, l[[h]]], {h, 1, Length[l]}], {t, 1, w}]/w][If[l == {}, 1, LCM @@ l]], If[i < 1, 0, Sum[b[n - i*j, i - 1, Join[l, Table[i, {j}]]]/j!/i^j, {j, 0, n/i}]]];
f[n_] := If[n == 0, 2, b[n, n, {}] - b[n - 1, n - 1, {}]]/2;
A055621 = f /@ Range[0, nmax];
mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
Join[{1}, EULERi[A055621 // Rest]] (* Jean-François Alcover, Jan 31 2020, after Alois P. Heinz in A055621 *)

Inverse Euler transform of A055621.

A317631 Number of connected set partitions of the vertices of labeled graphs with n vertices.

Original entry on oeis.org

1, 1, 1, 8, 200, 15901

Offset: 0

Gus Wiseman, Aug 02 2018

Gus Wiseman, All 200 connected set partitions of labeled graphs with 4 vertices.

Cf. A000110, A001187, A006125, A048143, A293510, A304717, A317632, A317634, A317635.

A317634 Number of caps (also clutter partitions) of clutters (connected antichains) spanning n vertices.

Original entry on oeis.org

1, 0, 1, 9, 315, 64880

Offset: 0

Gus Wiseman, Aug 02 2018

A kernel of a clutter is the restriction of the edge set to all edges contained within some connected vertex set. A clutter partition is a set partition of the edge set using kernels.

			The a(3) = 9 clutter partitions:
  {{{1,2,3}}}
  {{{1,3},{2,3}}}
  {{{1,2},{2,3}}}
  {{{1,2},{1,3}}}
  {{{1,3}},{{2,3}}}
  {{{1,2}},{{2,3}}}
  {{{1,2}},{{1,3}}}
  {{{1,2},{1,3},{2,3}}}
  {{{1,2}},{{1,3}},{{2,3}}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.
Gus Wiseman, All clutter partitions of non-isomorphic clutters on 4 vertices.

Cf. A001187, A030019, A048143, A275307, A286520,, A293510, A304717, A317631, A317632, A317635.

A317632 Number of connected induced nonempty non-singleton subgraphs of labeled connected graphs with n vertices.

Original entry on oeis.org

0, 0, 1, 13, 294, 12198, 946712, 140168924, 40223263760, 22598607583376, 24999757695984960, 54630901092648916704, 236304498092496715916416, 2026201628540583716863002880, 34482826679730591694177065948928, 1166004710785628820717860509317415168

Offset: 0

Gus Wiseman, Aug 02 2018

nonn

The edges of an induced subgraph G|S are those edges of G with both ends contained in S, where S is a subset of the vertices.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, All 294 connected induced subgraphs of labeled connected graphs with 4 vertices.

Cf. A001187, A006125, A048143, A293510, A304717, A317631, A317634, A317635.

seq(n)={
  my(p=sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n)));
  my(g=Vec(serlaplace(log(p))));
  my(q=sum(k=0, n, sum(j=2, k, binomial(k,j)*g[j]*2^(binomial(k-j, 2) + j*(k-j)))*x^k/k!, O(x*x^n)));
  Vec(serlaplace(q/p), -n-1)
} \\ Andrew Howroyd, Dec 10 2018

a(6) from Gus Wiseman, Dec 10 2018

Terms a(7) and beyond from Andrew Howroyd, Dec 10 2018

cp's OEIS Frontend

A323818 Number of connected set-systems covering n vertices.

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A007297 Number of connected graphs on n labeled nodes on a circle with straight-line edges that don't cross.

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A326754 BII-numbers of set-systems covering an initial interval of positive integers.

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A324173 Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.

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A309326 BII-numbers of minimal covers.

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A317672 Regular triangle where T(n,k) is the number of clutters (connected antichains) on n + 1 vertices with k maximal blobs (2-connected components).

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A323819 Number of non-isomorphic connected set-systems covering n vertices.

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A317631 Number of connected set partitions of the vertices of labeled graphs with n vertices.