A035053 -id:A035053 - cp's OEIS frontend

A007563 Number of rooted connected graphs where every block is a complete graph.

0, 1, 1, 3, 8, 25, 77, 258, 871, 3049, 10834, 39207, 143609, 532193, 1990163, 7503471, 28486071, 108809503, 417862340, 1612440612, 6248778642, 24309992576, 94905791606, 371691137827, 1459935388202, 5749666477454

Offset: 0

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 71, (3.4.13).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 0..1600 (first 200 terms from T. D. Noe)
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 167
N. J. A. Sloane, Transforms

Cf. A007549, A030019, A035051, A035052, A035053.
Column k=2 of A144042.
Cf. A245566.

with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): c:= etr(b): a:= n-> if n=0 then 0 else c(n-1) fi: seq(a(n), n=0..25); # Alois P. Heinz, Sep 06 2008

etr[p_] := etr[p] = Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a[0] = 0; a[n_] := etr[etr[a]][n-1]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, May 28 2013, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(v)))); concat([0], v)} \\ Andrew Howroyd, May 20 2018

Shifts left when Euler transform is applied twice.

a(n) ~ c * d^n / n^(3/2), where d = 4.189610958393826965527036454524044275... (see A245566), c = 0.1977574301782950818433893126632477845870281049591883888... . - Vaclav Kotesovec, Jul 26 2014

New description from Christian G. Bower, Oct 15 1998

A326752 BII-numbers of hypertrees.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 64, 128, 256, 260, 272, 276, 292, 304, 320, 512, 516, 532, 544, 548, 560, 576, 768, 784, 800, 1024, 1040, 1056, 2048, 2064, 2068, 2080, 2084, 2096, 2112, 2304, 2308, 2336, 2560, 2564, 2576, 2816, 3072, 4096, 4100, 4128, 4608

Offset: 1

Gus Wiseman, Jul 23 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. 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. In an antichain, no edge is a subset or superset of any other edge. A hypertree is a connected antichain of nonempty sets with density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices.

			The sequence of all hypertrees together with their BII-numbers begins:
    0: {}
    1: {{1}}
    2: {{2}}
    4: {{1,2}}
    8: {{3}}
   16: {{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  256: {{1,4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  292: {{1,2},{2,3},{1,4}}
  304: {{1,3},{2,3},{1,4}}
  320: {{1,2,3},{1,4}}

Cf. A000120, A000272, A030019 (spanning hypertrees), A035053, A048143, A048793, A052888, A070939, A134954, A275307, A326031, A326702, A326753.

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

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
density[c_]:=Total[(Length[#1]-1&)/@c]-Length[Union@@c];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Select[Range[0,1000],#==0||stableQ[bpe/@bpe[#],SubsetQ]&&Length[csm[bpe/@bpe[#]]]<=1&&density[bpe/@bpe[#]]==-1&]

A309314 BII-numbers of hyperforests.

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, 292, 304, 320, 512, 513, 516, 520, 521, 524, 528, 532

Offset: 1

Gus Wiseman, Jul 23 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. 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. In an antichain, no edge is a subset or superset of any other edge. A hyperforest is an antichain of nonempty sets whose connected components are hypertrees, meaning they have density -1, where density is the sum of sizes of the edges minus the number of edges minus the number of vertices.

			The sequence of all hyperforests 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}}
  130: {{2},{4}}
  131: {{1},{2},{4}}
  132: {{1,2},{4}}
  136: {{3},{4}}
  137: {{1},{3},{4}}

Cf. A000120, A030019, A035053, A048143, A048793, A052888, A070939, A134954, A275307, A326031, A326702, A326753.

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

A134956 Number of hyperforests with n labeled vertices: analog of A134954 when edges of size 1 are allowed (with no two equal edges).

Original entry on oeis.org

1, 2, 8, 64, 880, 17984, 495296, 17255424, 728771584, 36208782336, 2069977144320, 133869415030784, 9664049202221056, 770400218809384960, 67219977066339008512, 6372035504466437079040, 652103070162164448952320, 71656927837957783339925504

Offset: 0

Don Knuth, Jan 26 2008

nonn

			From _Gus Wiseman_, May 21 2018: (Start)
The a(2) = 8 hyperforests are the following:
  {{1},{2},{1,2}}
  {{1},{1,2}}
  {{2},{1,2}}
  {{1,2}}
  {{1},{2}}
  {{1}}
  {{2}}
  {}
(End)

D. E. Knuth: The Art of Computer Programming, Volume 4, Generating All Combinations and Partitions Fascicle 3, Section 7.2.1.4. Generating all partitions. Page 38, Algorithm H. - Washington Bomfim, Sep 25 2008

Alois P. Heinz, Table of n, a(n) for n = 0..335

Cf. A134958. - Washington Bomfim, Sep 25 2008

Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A144959, A304716, A304717, A304867, A304911.

with(combinat): p:= proc(n) option remember; add(stirling2(n-1, i) *n^(i-1), i=0..n-1) end: g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: a:= n-> `if`(n=0, 1, 2^n * g(n)): seq(a(n), n=0..30); # Alois P. Heinz, Oct 07 2008

p[n_] := p[n] = Sum[ StirlingS2[n-1, i]*n^(i-1), {i, 0, n-1}]; g[n_] := g[n] = p[n] + Sum[Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n-1}]; a[n_] := If[n == 0, 1, 2^n* g[n]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 13 2015, after Alois P. Heinz *)

Equals 2^n*A134954(n).

a(n) = Sum of n!prod_{k=1}^n\{ frac{ A134958(k)^{c_k} }{ k!^{c_k} c_k! } } over all the partitions of n, c_1 + 2c_2 + ... + nc_n; c_1, c_2, ..., c_n >= 0. - Washington Bomfim, Sep 25 2008

A034941 Number of labeled triangular cacti with 2n+1 nodes (n triangles).

Original entry on oeis.org

1, 1, 15, 735, 76545, 13835745, 3859590735, 1539272109375, 831766748637825, 585243816844111425, 520038240188935042575, 569585968715180280038175, 753960950911045074462890625, 1186626209895384011075327630625, 2190213762744801162239116550679375

Offset: 0

Christian G. Bower, Oct 15 1998

nonn

Also the number of 3-uniform hypertrees spanning 2n + 1 labeled vertices. - Gus Wiseman, Jan 12 2019

Number of rank n+1 simple series-parallel matroids on [2n+1]. - Matt Larson, Mar 06 2023

			a(3) = 5!! * 7^2 = (1*3*5) * 49 = 735.
From _Gus Wiseman_, Jan 12 2019: (Start)
The a(2) = 15 3-uniform hypertrees:
  {{1,2,3},{1,4,5}}
  {{1,2,3},{2,4,5}}
  {{1,2,3},{3,4,5}}
  {{1,2,4},{1,3,5}}
  {{1,2,4},{2,3,5}}
  {{1,2,4},{3,4,5}}
  {{1,2,5},{1,3,4}}
  {{1,2,5},{2,3,4}}
  {{1,2,5},{3,4,5}}
  {{1,3,4},{2,3,5}}
  {{1,3,4},{2,4,5}}
  {{1,3,5},{2,3,4}}
  {{1,3,5},{2,4,5}}
  {{1,4,5},{2,3,4}}
  {{1,4,5},{2,3,5}}
The following are non-isomorphic representatives of the 2 unlabeled 3-uniform hypertrees spanning 7 vertices, and their multiplicities in the labeled case, which add up to a(3) = 735:
  105 X {{1,2,7},{3,4,7},{5,6,7}}
  630 X {{1,2,6},{3,4,7},{5,6,7}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
Maryam Bahrani and Jérémie Lumbroso, Enumerations, Forbidden Subgraph Characterizations, and the Split-Decomposition, arXiv:1608.01465 [math.CO], 2016.
Luis Ferroni and Matt Larson, Kazhdan-Lusztig polynomials of braid matroids, arXiv:2303.02253 [math.CO], 2023.
Katie Gedeon, N. Proudfoot, and B. Young, Kazhdan-Lusztig polynomials of matroids: a survey of results and conjectures, arXiv preprint arXiv:1611.07474 [math.CO], 2016-2017.
Nicholas Proudfoot and Ben Young, Configuration spaces, FS^op-modules, and Kazhdan-Lusztig polynomials of braid matroids, arXiv:1704.04510 [math.RT], 2017.
Eric Weisstein's World of Mathematics, Cactus Graph
Index entries for sequences related to cacti

Cf. A000272, A000665, A003081, A030019, A035053, A125791, A302374, A320444, A323292, A323298.

[(2*n+1)^(n-1)*Factorial(2*n)/(2^n*Factorial(n)): n in [0..15]]; // Vincenzo Librandi, Feb 19 2020

Table[(2n+1)^(n-1)(2n)!/(2^n n!), {n, 0, 14}] (* Jean-François Alcover, Nov 06 2018 *)

a(n) = A034940(n)/(2n+1).

The closed form a(n) = (2n-1)!! (2n+1)^(n-1) can be obtained from the generating function in A034940. - Noam D. Elkies, Dec 16 2002

Typo in a(10) corrected and more terms from Alois P. Heinz, Jun 23 2017

A007549 Number of increasing rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

1, 1, 3, 14, 89, 716, 6967, 79524, 1041541, 15393100, 253377811, 4596600004, 91112351537, 1959073928124, 45414287553455, 1129046241331316, 29965290866974493, 845605519848379436, 25282324544244718411, 798348403914242674980, 26549922456617388029641

Offset: 1

N. J. A. Sloane

In an increasing rooted graph, nodes are numbered and the numbers increase as you move away from the root.

(a(n+1)/a(n))/n tends to 1/A073003 = 1.676875... (same limit as A029768). - Vaclav Kotesovec, Jul 26 2014

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

Vaclav Kotesovec, Table of n, a(n) for n = 1..410 (first 200 terms from Vincenzo Librandi)
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

Cf. A007563, A030019, A035051-A035053.
Cf. A029768.
Row sums of A078341. Column k=1 of A264436.

exptr:= proc(p) local g; g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: end: b:= exptr(exptr(a)): a:= n-> `if`(n=0, 1, b(n-1)): seq(a(n), n=1..30); # Alois P. Heinz, Oct 07 2008

exptr[p_] := Module[{g}, g[n_] := g[n] = p[n] + Sum[ Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n-1}]; g]; b = exptr[ exptr[a] ]; a[n_] := If[n == 0, 1, b[n-1]]; Table[ a[n], {n, 1, 19}] (* Jean-François Alcover, May 10 2012, after Alois P. Heinz *)

Shifts left when exponentiated twice.
Conjecture: a(n) = Sum_{i=0..2^(n-2) - 1} b(i) for n > 1 with a(1) = 1 where b(n) = (L(n) + 2)*b(f(n)) + Sum_{k=0..L(n) - 1} (1 - R(n,k))*b(f(n) + 2^k*(1 - R(n,k))) for n > 0 with b(0) = 1, L(n) = A000523(n), f(n) = A053645(n) and where R(n,k) = floor(n/2^k) mod 2. Here R(n,k) is the (k+1)-th bit from the right side in the binary expansion of n. - Mikhail Kurkov, Jul 21 2024
Conjecture: a(n) = D^(n-1)(exp(x)) evaluated at x = 0, where D denotes the operator exp(x)*(1 + x)*d/dx. - Peter Bala, Feb 24 2025

New description from Christian G. Bower, Oct 15 1998

A304386 Number of unlabeled hypertrees (connected antichains with no cycles) spanning up to n vertices and allowing singleton edges.

Original entry on oeis.org

1, 2, 5, 15, 50, 200, 907, 4607, 25077, 144337, 863678, 5329994, 33697112, 217317986, 1424880997, 9474795661, 63769947778, 433751273356, 2977769238994, 20611559781972, 143720352656500, 1008765712435162, 7122806053951140, 50566532826530292, 360761703055959592

Offset: 0

Gus Wiseman, May 21 2018

nonn

			Non-isomorphic representatives of the a(3) = 15 hypertrees are the following:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{2},{1,2}}
  {{1,3},{2,3}}
  {{3},{1,2,3}}
  {{1},{2},{1,2}}
  {{3},{1,2},{2,3}}
  {{3},{1,3},{2,3}}
  {{2},{3},{1,2,3}}
  {{1},{2},{3},{1,2,3}}
  {{2},{3},{1,2},{1,3}}
  {{2},{3},{1,3},{2,3}}
  {{1},{2},{3},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A030019, A035053, A048143, A054921, A134954, A134955, A134957, A134959, A144959, A304386, A304717, A304867, A304911, A304912, A304968, A304970.

\\ here b(n) is A318494 as vector
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
b(n)={my(v=[1]); for(i=2, n, v=concat([1], EulerT(EulerT(2*v)))); v}
seq(n)={my(u=2*b(n)); Vec(1 + x*Ser(EulerT(u))*(1-x*Ser(u))/(1-x))} \\ Andrew Howroyd, Aug 27 2018

Partial sums of b(1) = 1, b(n) = A134959(n) otherwise.

Terms a(7) and beyond from Andrew Howroyd, Aug 27 2018

A320444 Number of uniform hypertrees spanning n vertices.

Original entry on oeis.org

1, 1, 1, 4, 17, 141, 1297, 17683, 262145, 4861405, 100112001, 2371816701, 61917364225, 1796326510993, 56693912375297, 1947734359001551, 72059082110369793, 2863257607266475419, 121439531096594251777, 5480987217944109919765, 262144000000000000000001

Offset: 0

Gus Wiseman, Jan 09 2019

nonn

The density of a hypergraph is the sum of sizes of its edges minus the number of edges minus the number of vertices. A hypertree is a connected hypergraph of density -1. A hypergraph is uniform if its edges all have the same size. The span of a hypergraph is the union of its edges.

			Non-isomorphic representatives of the 5 unlabeled uniform hypertrees on 5 vertices and their multiplicities in the labeled case, which add up to a(5) = 141:
   5 X {{1,5},{2,5},{3,5},{4,5}}
  60 X {{1,4},{2,5},{3,5},{4,5}}
  60 X {{1,3},{2,4},{3,5},{4,5}}
  15 X {{1,2,5},{3,4,5}}
   1 X {{1,2,3,4,5}}

Robert Israel, Table of n, a(n) for n = 0..387

Row sums of A326374.

Cf. A000272, A030019, A035053, A038041, A052888, A057625, A061095, A121860, A134954, A236696, A262843.

f:= proc(n) local d; add((n-1)!/(d! * ((n-1)/d)!) * (n/d)^((n-1)/d - 1), d = numtheory:-divisors(n-1)); end proc:
f(0):= 1: f(1):= 1:
map(f, [$0..25]); # Robert Israel, Jan 10 2019

Table[Sum[n!/(d!*(n/d)!)*((n+1)/d)^(n/d-1),{d,Divisors[n]}],{n,10}]

a(n) = if (n<2, 1, n--; sumdiv(n, d, n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1))); \\ Michel Marcus, Jan 10 2019

a(n + 1) = Sum_{d|n} n!/(d! * (n/d)!) * ((n + 1)/d)^(n/d - 1).

a(p prime) = 1 + (p + 1)^(p - 1).

A304918 Number of labeled antichain hyperforests spanning a subset of {1,...,n}.

Original entry on oeis.org

1, 2, 5, 18, 104, 943, 12133, 203038, 4177755, 101922814, 2874725600, 92009680557, 3294276613933, 130446181101044, 5660055256165565, 267044522107706072, 13611243187516647324, 745329728016955480687, 43636132793651444511809, 2719977663069107176768790

Offset: 0

Gus Wiseman, May 21 2018

nonn

			The a(3) = 18 hyperforests are the following:
{{1,2,3}}      {{2,3}}    {{1,3}}    {{1,2}}    {{3}}   {{2}}   {{1}}   {}
{{1,3},{2,3}}  {{2},{3}}  {{1},{3}}  {{1},{2}}
{{1,2},{2,3}}
{{1,2},{1,3}}
{{3},{1,2}}
{{2},{1,3}}
{{1},{2,3}}
{{1},{2},{3}}

Cf. A030019, A035053, A048143, A134954, A134955, A134957, A134956, A134957, A134959, A144959, A303838, A304716, A304717, A304867, A304911, A304912.

Binomial transform of A134954.

A035052 Number of sets of rooted connected graphs where every block is a complete graph.

Original entry on oeis.org

1, 1, 2, 5, 14, 42, 134, 444, 1518, 5318, 18989, 68856, 252901, 938847, 3517082, 13278844, 50475876, 193014868, 741963015, 2865552848, 11113696421, 43266626430, 169019868095, 662337418989, 2602923589451, 10256100717875

Offset: 0

Christian G. Bower, Oct 15 1998

nonn

T. D. Noe, Table of n, a(n) for n=0..200
Loic Foissy, The Hopf algebra of Fliess operators and its dual pre-Lie algebra, 2013.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 862
N. J. A. Sloane, Transforms

Cf. A007549, A007563, A030019, A035051, A035053.

Cf. A245566.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0,1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: b:= etr(aa): c:= etr(b): aa:= n-> if n=0 then 0 else c(n-1) fi: a:= etr(aa): seq(a(n), n=0..25); # Alois P. Heinz, Sep 09 2008

etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[ j]}]*b[n-j], {j, 1, n}]/n]; b]; b = etr[aa]; c = etr[b]; aa = Function[{n}, If[n == 0, 0, c[n-1]]]; a = etr[aa]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]);for(i=2, n, v=concat([1], EulerT(EulerT(v)))); concat([1], EulerT(v))} \\ Andrew Howroyd, May 20 2018

Euler transform of A007563.

a(n) ~ c * d^n / n^(3/2), where d = 4.189610958393826965527036454524... (see A245566), c = 0.35683683547585... . - Vaclav Kotesovec, Jul 26 2014

cp's OEIS Frontend

A007563 Number of rooted connected graphs where every block is a complete graph.

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A326752 BII-numbers of hypertrees.

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Mathematica

A309314 BII-numbers of hyperforests.

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A134956 Number of hyperforests with n labeled vertices: analog of A134954 when edges of size 1 are allowed (with no two equal edges).

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A034941 Number of labeled triangular cacti with 2n+1 nodes (n triangles).

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A007549 Number of increasing rooted connected graphs where every block is a complete graph.

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A304386 Number of unlabeled hypertrees (connected antichains with no cycles) spanning up to n vertices and allowing singleton edges.

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A320444 Number of uniform hypertrees spanning n vertices.